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ARITHMETIC 


UPON    THE 


INDUCTIVE    METHOD    OF    INSTRUCTION: 


A   SEQUEL 


INTELLECTUAL  ARITHMETIC, 


BY  WARREN  COLBURN,  A.  M. 


NEW-YORK: 

PUBLISHED   BY    ROE    LOCKWOOD. 

BOSTON: 
HILLIARD,  GRAY  AND  CO. 

1833. 


DISTRICT  OF  MASSACHUSETTS,  to  -wit  . 

District  Clerk's  Office, 

BE  IT  REMEMBERED,  that  on  the  twenty-fifth  day  of  May,  A.  D 
1826,  and  in  the  fiftieth  year  of  the  Independence  of  the  United  States 
of  America,  Warren  Colburn,  of  the  said  district,  has  deposited  in 
this  office  the  title  of  a  book,  the  right  whereof  he  claims  as  author,  in 
the  words   following,  to  wit : 

"  Arithmetic  upon  the  Inductive  Method  of  Instruction  :  being  a 
Sequel  to  Intellectual  Arithmetic.     By  Warren  Colburn,  A.  M." 

In  conformity  to  the  Act  of  the  Congress  of  the  United  States,  en- 
titled, *'  An  act  for  the  encouragement  of  learning,  by  securing  the 
copies  of  maps,  charts,  and  books,  to  the  authors  and  proprietors  of 
such  copies,  during  the  times  therein  mentioned  ;"  and  also  an  act,  en 
titled,  ''  An  act  supplementary  to  an  act,  entitled,  An  act  for  the  en- 
coura-gement  of  learning,  by  securing  the  copies  of  maps,  charts,  and 
books"  to  the  authors  and  proprietors  of  such  copies,  during  the  times 
therein  mentioned ;  and  extending  the  benefits  thereof  to  the  arts  of 
designincr,  engraving,  and  etching,  historical  and  other  prints  " 

JNO.  W.  DAVIS, 
CUrk  qf  the  District  of  Massachusetts. 


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RECOMMENDATIONS.   ^    5'? 


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From  B.  A.  Gould,  Principal  of  the  Public  Latin 'School,  Boston 

Boston^  22d  Oct.,  1822. 
,  Dear  Sir, 

I  have  been  highly  gratified  by  the  examination  of  the  second 
part  of  your  Arithmetic.  The  principles  of  the  science  are  unfolded, 
and  its  practical  uses  explained  with  great  perspicuity  and  simpUcity. 
1  think  your  reasonings  and  illustrations  are  peculiarly  happy  and 
original.  This,  together  with  your  "  First  Lessons,"  forms  the  most 
lucid  and  intelligible,  as  well  as  the  most  scientific  system  of  Arith- 
metic I  have  ever  seen. — Its  own  merits  place  it  beyond  the  need  of 
commendation. 

With  much  esteem. 

Sir,  your  obedient  servant, 

B.  A.  GOULD. 
Mr.  Warren  Colburn. 


From  G.  B.  Emersoit,  Principal  of  the  English  Classical  School, 
Boston. 

Boston,  22d  Oct.,  1822, 
Dear  Sir, 

I  have  carefully  examined  a  large  portion  of  your  manuscript, 
and  do  not  hesitate  to  recommend  it  very  highly  to  every  person  who 
wishes  to  teach  arithmetic  intelligibly.  The  arrangement  is  very 
much  better,  the  explanations  more  convincing,  and  the  rules,  from 
the  mode  in  which  they  are  introduced,  are  cleajer  and  simpler,  than 
can  be  found  in  any  book  on  the  subject  with  which  I  am  acquainted. 
I  am,  with  great  respect, 

Yours,  &c. 

G.  B.  EMERSON. 
Mr.  Warren  CoLBaair. 


M'^syeso 


PREFACE 


It  will  be  «xtremely  useful,  though  not  absolutely  necessary,  foi 
pupils  of  every  age  to  study  the  "  First  Lessons,"  previous  to  com- 
mencing this  tieatise.  There  is  an  intimate  connexion  between  the 
two,  though  this  is  not  dependent  on  the  other.  It  is  hoped  that  this 
will  b9  found  less  difficult  than  other  treatises  on  the  subject,  for  those 
who  have  not  studied  the  "  First  Lessons." 

Pupils  may  commence  the  "  First  Lessons"  to  advantage,  as  soon  as 
they  can  read  the  examples ;  and  even  before  they  can  read,  it  will 
be  found  very  useful  to  ask  them  questions  from  it.  This  may  be  done 
by  other  pupils  who  have  already  studied  it.  Those  who  commence 
early,  may  generally  obtain  sufficient  knowledge  of  it  by  the  time  they 
are  eight  or  nine  years  old.     They  may  then  commence  this. 

This  Sequel  consisU  of  two  parts.  The  first  contains  a  course  of 
examples  for  the  illustration  and  application  of  the  principles.  The 
second  part  contains  a  developeraent  of  the  principles.  The  articles 
are  numbered  in  the  two,  so  as  to  co-respond  with  each  other.  The 
two  parts  are  to  be  studied  together,  when  the  pupil  is  old  enough  to 
comprehend  the  second  part  by  reading  it  himself  When  he  has 
performed  all  tne  examples  in  an  article  in  the  first  part,  he  should  be 
required  to  recite  the  oorrespondmg  article  in  the  second  part,  not 
verbatim,  but  to  give  a  ^ood  account  of  the  reasoning.  When  the 
principle  is  well  understood,  the  rules  which  are  printed  in  Italics 
should  be  committed  to  memory.  At  each  recitation,  the  first  thing 
should  be  to  require  the  pupil  to  give  a  practical  example,  invohing 
the  principle  to  be  explained,  and  then  an  explanation  of  the  principle 
itself. 

When  the  pupil  is  to  karn  the  use  of  figures  for  the  first  time,  it  is 
best  to  explain  to  him  the  nature  of  them  as  in  Art.  I.,  to  about  three 
or  four  places  ;  and  then  require  him  to  write  some  numbers.  Thei» 
give  him  some  of  the  first  examples  in  Art.  II.,  without  telling  him 
what  to  do.  He  will  discover  what  is  to  be  di>ne,  and  invent  a  way 
lo  do  it.  Let  him  perform  several  in  his  own  way,  and  then  suggest 
some  method  a  little  different    from  his,    and  nearer  the    common 


PREFACE.  O 

method.  If  he  readily  comprehends  it,  he  will  be  pleased  with  it, 
and  adopt  it.  If  he  does  not,  his  mind  is  not  yet  prepared  for  it,  and 
should  be  allowed  to  continue  his  own  way  longer,  and  then  it  should 
be  suggested  again.  After  he  is  familiar  with  that,  suggest  another 
method,  somewhat  nearer  the  common  method,  and  so  on,  until  he 
learns  the  best  method.  Never  urge  him  to  adopt  any  method  until 
he  understands  it,  and  is  pleased  with  it.  In  some  of  the  articles,  it 
may  perhaps  be  necessary  for  young  pupils  to  perform  more  examples 
than  are  given  in  the  book. 

When  the  pupil  is  to  commence  multiplication,  give  him  one  of  the 
first  examples  in  Art.  III.,  as  if  it  were  an  example  in  Addition.  He 
will  write  it  down  as  such  But  if  he  is  familiar  with  the  "  First 
Lessons,"  he  will  probably  perform  it  as  multiplication  without  know- 
ing it.  When  he  does  this,  suggest  to  him,  that  he  need  not  write  the 
number  but  once.  Afterwards  recommend  to  him  to  write  a  number, 
to  show  how  many  times  he  repeated  it,  lest  he  should  forget  it. 
Then  tell  him  that  it  is  Multiplication.  Proceed  in  a  similar  manner 
with  the  other  rules. 

One  general  maxim  to  be  observed  with  pupils  of  every  age,  is 
never  to  tell  them  directly  how  to  perform  any  example.  If  a  pupil 
is  unable  to  perform  an  example,  it  is  generally  because  he  does  not 
fully  comprehend  the  object  of  it.  The  object  should  be  explained, 
and  some  questions  asked,  which  will  have  a  tendency  to  recal  the 
principles  necessary.  If  this  does  not  succeed,  his  mind  is  not  pre- 
pared for  it,  and  he  must  be  required  to  examine  it  more  by  hinibelf, 
and  to  review  some  of  the  principles  which  it  involves.  It  is  useless 
for  him  to  perform  it  before  his  mind  is  prepared  for  it.  After  he  has 
been  told,  he  is  satisfied,  and  will  not  be  willing  to  examine  the 
principle,  and  he  will  be  no  better  prepared  for  another  case  of  the 
same  kind,  than  he  was  before.  When  the  pupil  knows  that  he  is  not 
to  be  told,  he  learns  to  depend  on  himself;  and  when  he  once  con- 
tracts the  habit  of  understanding  what  he  does,  he  will  not  easily  be 
prevailed  on  to  do  any  thing  which  he  does  not  understand. 

Several  considerations  induce  the  author  to  think,  that  when  a 
principle  is  to  be  taught,  practical  questions  should  first  be  proposed, 
care  being  taken  to  select  such  as  will  show  the  combination  in  the 
simplest  manner,  and  that  the  numbers  be  so  smaL  that  the  operation 
shall  not  be  difficult.  When  a  proper  idea  is  formed  of  the  nature 
and  use  of  the  combination,  the  method  of  solving  these  questions 
with  large  numbers  should  be  attended  to.  This  metnod,  on  trial 
has  succeeded  beyond  his  expectations.  Practical  examples  not  only 
show  at  once  the  object  to  be  accomplished,  but  they  greatlv  assist 
1  * 


G  PREFACE. 

the  imagination  in  unfolding  the  principle  and  discovering  the  opera- 
tions requisite  for  the  solution. 

This  principle  is  mado  the  basis  of  this  treatise  ;  viz.  whenever  a 
new  combination  is  infroduced,  it  is  done  with  practical  examples, 
proposed  in  such  a  manner  as  to  show  what  it  is,  ami  as  much  as 
possible,  how  it  is  to  be  performed.  The  examples  are  so  small  that 
the  pupil  may  easily  reason  upon  them,  and  that  there  will  be  no 
difficulty  in  the  operation  itself,  until  the  combination  is  well  under- 
stood. In  this  way  it  is  believed  that  the  leading  idea  which  the 
pupil  will  obtain  of  each  combination,  \vill  be  the  effect  which  will  be 
produced  by  it,  rather  than  how  to  perform  it,  though  the  latter  will 
be  sufficiently  well  understood. 

The  second  part  contains  an  analytical  developement  of  the  princi- 
ples. Almost  all  the  examples  used  for  this  purpose  are  practical. 
Care  has  been  taken  to  make  every  principle  depend  as  little  as 
possible  upon  others.  Young  persons  cannot  well  follow  a  course  of 
reasoning  where  one  principle  is  built  upon  another.  Besides,  a  prin- 
ciple is  always  less  understood  by  every  one,  in  proportion  as  it  is 
made  to  depend  on  others. 

In  tracing  the  principles,  several  distinctions  have  been  made  which 
have  not  generally  been  made.  They  are  principally  in  division  of 
whole  numbers,  and  in  division  of  whole  numbers  by  fractions,  and 
fractions  by  fractions.  There  are  some  instances  also  of  combinations 
being  classed  together,  which  others  have  kept  separate. 

As  the  purpose  is  to  give  the  learner  a  knowledge  of  the  principles, 
it  is  necessary  to  have  the  variety  of  examples  under  each  principle 
as  great  as  possible.  The  usual  method  of  arrangement,  according  to 
subjects,  has  been  on  this  account  entirely  rejected,  and  the  arrange- 
ment has  been  made  according  to  principles.  Many  different  subjects 
come  under  tho  same  principle  ;  and  different  parts  of  the  same  sub- 
ject frequently  come  under  different  principles.  When  the  principles 
are  well  understood,  very  few  subjects  will  require  a  particular  rule, 
and  if  the  pupil  is  properly  introduced  to  them,  he  will  understand 
them  better  without  a  rule  than  with  one.  Besides,  he  will  be  better 
prepared  for  the  cases  which  occur  in  business,  as  he  will  be  obliged 
to  meet  them  there  without  a  name.  The  different  subjects,  as  they 
are  generally  arranged,  often  embarrass  the  learner.  When  he  meets 
with  a  name  with  whicn  he  is  not  acquainted,  and  a  rule  attached  to 
it,  he  is  frequently  at  a  loss,  when  if  he  saw  the  example  without  the 
name,  he  would  not  hesitate  at  all. 

The  manner  of  performing  examples  will  appear  new  to  mr^ny,  but 
it  will  be  found  much  more  agreeable  to  the  practice  of  men  of  busi- 


PRKFAUE.  7 

Hess,  and  men  of  science  generally,  than  those  commonly  found  in 
books.  This  is  the  method  of  those  that  understand  the  subject.  The 
others  were  invented  as  a  substitute  for  understanding. 

The  rule  of  three  is  entirely  omitted.  This  has  been  considered 
useless  in  France,  for  some  yeais,  though  it  has  been  retained  in  their 
books.  Those  who  understand  the  principles  sufficiently  to  compre- 
hend the  nature  of  the  rule  of  three,  can  do  much  better  without  it 
than  with  it,  for  when  it  is  used,  it  obscures,  rather  than  illustrates, 
the  subject  to  which  it  is  applied.  The  principle  of  the  rale  of  three 
is  similar  to  the  combinations  in  Art.  XVI. 

The  rule  of  Position  has  been  omitted.  This  is  an  artificial  rule, 
the  principle  of  which  cannot  be  well  understood  without  the  aid  of 
Algebra :  and  when  Algebra  is  understood,  Position  is  useless.  Be- 
sides, all  the  examples  which  can  be  performed  by  Position,  may  be 
performed  much  more  easily,  and  in  a  manner  perfectly  intelligible, 
without  it.  The  manner  in  which  they  are  performed  is  similar  to 
that  of  Algebra,  but  without  Algebraic  notation.  The  principle  of 
false  position,  properly  so  called,  is  applied  only  to  questions  where 
there  are  not  sufficient  data  to  solve  them  directly. 

Powers  and  roots,  though  arithmetical  operations,  come  more  pro- 
perly within  the  province  of  Algebra. 

There  are  no  answers  to  the  examples  given  in  the  book.  A  key  is 
published  separately  for  teachers,  containing  the  answers  and  solutions 
of  the  most  difficult  examples. 


TABLE  OF  CONTENTS. 


(This  Table  equally  refers  to  Parts  I.  and  11.) 


I.  Numeration  and  Notation. 

II.  Addition. 

III.  Multiplication,  when  the  multiplier  is  a  single  figure. 

IV.  Compound  numbers,  factors,  and  multiplication,  when  the  multi- 

plier is  a  compound  number. 

V.  Multiplication,  when  the  multiplier  is  10,  100,  1000,  «&c. 

VI.  Do.  when  the  multiplier  is  20,  300,  &c. 

VII.  Do.  when  the    multiplier   consists  of  any  number  of 
figures. 

VIII.  Subtraction. 

IX.  Division,  to  find  how  many  times  one  number  is  contained  in 

another. 

X.  Division.     Explanation  of  Fractions.     Their  Notation-     What  is 

to  be  done  with  the  remainder  after  division. 

XI.  Division,  when  the  divisor  is  10,  100,  &;c. 

XII.  To  find  what  part  of  one  number  another  is,  or  to  find  the  ratio 

of  one  number  to  another. 

XIII.  To  change  an  improper  fraction  to  a  whole  or  mixed  number. 

XIV.  To  change  a  whole  or  mixed  number  to  an  improper  fraction. 

XV.  To  multiply  a  fraction  by  a  whole  number,  by  multiplying  the 

numerator. 

XVI.  Division,  to  divide  a  number  into  parts.     To  multiply  a  whole 

number  by  a  fraction. 

XVII.  To  divide  a  fraction  by  a  whole  number.     To  multiply  a  frac- 
tion by  a  fraction. 

XVIII.  To  multiply  a  fraction  by  dividing  the  denominator.     Two 
ways  to  multiply,  and  two  ways  to  divide,  a  fraction. 

XIX.  Addition  and  subtraction  of  fractions.     To  reduce  them  to 
common  denominator.     To  reduce  them  to  lower  terms 


CONTENTS.  9 

XX.  Contractions  in  division. 

XXI.  How  to  find  the  divisors  of  numbers.  To  find  the  greatest  com* 
mon  divisor  of  two  or  more  numbers  To  reduce  fractions  to 
their  lowest  terms. 

XXII.  To  find  the  least  oommoA  multiple  of  two  or  mot-e  numbers. 
To  reduce  fractions  to  the  least  common  denominator. 

XXIII.  To  divide  a  whole  number  by  a  fraction,  or  a  fraction  by  a 
fiaction,  when  the  purpose  is  to  find  how  many  times  the  divi- 
sor is  contained  in  the  dividend.  To  find  the  rt^tiQ  iof  a  fraction 
and  a  whole  number,  or  of  two  fractions. 

XXIV.  To  divide  a  whole  number  by  a  fraction,  or  a  fraction  by  a 
fraction ;  a  part  of  a  number  being  given  to  find  the  whole. 
This  is  on  the  same  principle  as  that  of  dividing  a  number  into 
parts. 

XXV.  Decimal  Fractions.     Numeration  and  notation  of  them. 

XXVI.  Addition  and  Subtraction  of  Decimals.  To  change  a  common 
fraction  to  a  decimal. 

XXVII.  Multiplication  of  Decimals. 

XXVIII.  Division  of  Decimals. 

XXIX.  Circulating  Decimals. 

Proof  of  multiplication  and  division  by  casting  out  9  > 


INDEX  TO  PARTICULAR  SUBJECTS 


Compound  Multiplication  )                                            Page.  Example 

Addition          >  Miseellaneoua  examples         37  1....49 
Subtraction      ) 

Division             Miscellaneous  eramples       211  1....25 

Interest,  Simple  ^ 

Commission                                                                         (    28  43... .50 

Insurance                        • <    92  65. .113 

Duties  and  Premiums                                                      (  104  43....74 
Discount,  Common      J 

Compound  Interest •  •  *    215  58....63 

T.           ,                                                                       5    78  130..142 

^^^^^°"' i  224  110..113 

„  ^                                                                                   5    36  102..106 

^^^^^ \    42  34....38 

,             ,  ^  .                                                                    J  103  33....41 

Loss  and  Gam <  gj^  ^g..  .57 

V  u       w     a-      1                                                           5    58  158..J66 

Fellowship,  Simple <  goQ  65    86 

Fellowship,  Compound 221  87....92 

Equation  of  Payments 222  103..109 

Alligation  Medial 218  69....72 

Alligation  Alternate 218  73....84 

C    79  1....49 

Square  and  Cubic  Measure.  Miscellaneous  Examples  <    91  56... .64 

( 101  13....26 

Duodecimals 229  141..144 

Taies 103  28....32 

Measure  of  circles,  parallelograms,  triangles,  &c.     .   .233  181. .187 

Geographical  and  Astronomical  questiuns 234  188..198 

Exchange 235  199..205 

Tables  of  Coin,  Weights,  and  Measures 236 

Reflections  on  Mathematical  reasoning 240 


ARITHMETIC. 


PART  I. 


ADDITION. 

The  student  may  perform  the  followmg  examples  Ji  his 
mind. 

1.  James  has  3  cents  and  Charles  has  5  ;  how  many  hare 
they  both  ? 

2.  Charles  bought  3  buna«  for  16  cents,  a  quart  of  cher- 
ries for  8  cents,  and  2  oranges  for  12  cents  ;  how  many  cents 
did  he  lay  out  1 

3.  A  man  bought  a  hat  for  8  dollars,  a  coat  for  27  dollars, 
a  pair  of  boots  for  5  doHars,  and  a  vest  for  7  dollars ;  how 
many  dollars  did  the  whole  come  to  ? 

4.  A  man  bought  a  firkin  of  butter  for  8  dollars,  a  quarter 
of  veal  for  45  cents,  and  a  barrel  of  cider  for  3  dollars  and  25 
cents  ;   how  much  did  he  give  for  the  whole  ? 

5.  A  man  sold  a  horse  for  127  dollars,  a  load  of  hay  for  15 
dollars,  and  3  barrels  of  cider  for  12  dollars ;  how  much  did 
he  receive  for  the  whole  ? 

6.  A  man  travelled  27  miles  in  one  day,  15  miles  the  next 
day,  and  8  miles  the  next ;  how  many  miles  did  he  travel  in 
the  whole  ? 

7.  A  man  received  42  dollars  and  37  cents  of  one  person, 
4  dollars  and  68  cents  of  another,  and  7  dollars  and  83  cents 
of  a  third ;  how  much  did  he  receive  in  the  whole  ? 

8.  I  received  25  dollars  and  58  cents  of  one  man,  45  dol- 
lars and  83  cents  of  another,  and  8  dollars  and  39  cents  of  a 
third ;  how  much  did  I  receive  in  the  whole  ? 

The  two  last  examples  may  be  performed  in  the  mind,  but 
they  will  be  rather  difficult.  A  more  convenient  method 
will  soon  be  found. 


12 


ARITHMETIC. 


Parti. 


NUMERATION. 
1.  Write  in  words  the  following  numbers. 

1  27 

S  35 

3  58 

4  63 

5  70 

6  84 

7  96 

8  100 

9  103 

10  110 

11  113 

12  127 

13  308 

14  520 

15  738 

16  1,000 

17  1,001 

18  1,010 

19  1,100 

20  1,018 

21  2,107 

22  3  250 

23  5,796 

Write  in  figures  the  following  numbers. 
J.  Thirty-four. 

2.  Fifty-seven. 

3.  Sixty-three. 

4.  Eighty. 

5.  One  hundred. 

6.  One  hundred  and  one. 

7.  One  hundred  and  ten. 

8.  Three  hundred  and  eleven. 

9.  Five  hundred  and  seventeen. 

10.  Eight  hundred  and  fifty. 

1 1.  Nine  hundred  and  eighty-six. 

12.  One  thousand  and  one. 

13.  One  thousand  and  ten. 

14.  Three  thousand,  one  hundred  and  one. 

15.  Five  thousand  and  sixty. 


24 

10,000 

25 

20,030 

26 

50,705 

27 

67,083 

28 

300,050 

29 

476,089 

30 

707,720 

31 

1,000,370 

32 

5,600,073 

33 

8,081,305 

34 

59,006,341 

35 

305,870,400 

36 

590,047,608 

37 

1,000,000,000 

38 

3,670,000,387 

39 

45,007,070,007 

40 

680,930,100,700 

41 

50,787,657,000,500 

42 

270,000,838,003,908 

43 

68,907,605 

44 

56,000,034,750 

45 

6,703,720,000.857 

ir.  ADDITTON.  13 

16.  Ten  thousand  and  five. 

17.  Thirty  thousand,  five  hundred,  and  four. 

18.  Sixty-seven  thousand,  and  forty. 

19.  Five  hundred  thousand,  and  seventy-one. 

20.  Two  hundred  and  seven  thousand,  six  hundred. 

21.  Four  milHons,  sixty  thousand,  and  eighty-four. 

22.  Ninety-seven  millions,  thirty-five  thousand,  eight  hun- 
dred and  five. 

23.  Fifty  millions,  s^^venty  thousand,  and  eight. 

24.  Three  hundred  millions,  and  fifty-seven. 

25.  Two  billions,  fifty-three  millions,  three  hundred  and 
five  thousand,  two  hundred. 

26.  Fifty  billions,  two  hundred  and  seven  millions,  sixty- 
seven  thousand,  two  hnaJrcd. 

27.  Eighty-seven  millions,  and  sixty-three. 

28.  Six  hundred  billions,  two  hundred  and  seven  thousand, 
and  three. 

29.  Thirty-five  trilFions,  nine  millions,  and  fifty-eight. 

30.  Six  hundred  and  fifty-seven  trillions,  seven  billions, 
ninety-seven  thousand,  and  sixty-seven. 

31.  Seventy  miTiions,    two  hundred   nnd    fifty  thousand, 
three  hundred  and  sixty-seven. 

32.  Four  hundred  and  seven  trillions,  and  eighty-seven 
thousand. 

33.  Thirty-'five  billions,  ninety-eight  thousand,  one  hun- 
dred. 

34.  Forty   millions,  two  hundred  thousand,  and  seventy- 
four. 

35.  Eighty-three  millions,  seven  hundred  and  sixty-three 
thousand,  nine  hundred  and  fifty-seven. 


ADDITION. 

II.  1.*  A  man  bought  a  watch  for  fifty-eight  dollars,  a 
cane  for  five  dollars,  a  hat  for  ten  dollars,  and  a  pair  of  boost 
for  six  dollars.     What  did  he  give  for  the  whole  ? 

2.  In  an  orchard  there  are  six  rows  of  trees  ;  in  the  two 
first  rows,  there  are  fifteen  trees  in  each  row ;  in  the  third 
row,  seventeen  ;  in  the  fourth  row,  eleven ;  in  the  fifth  row, 

•*  See  First  Lessons,  sect.  I. 
2 


U  ARlTIIMETiC.  Part  1 

eight ;  and  in  the  sixth  row,  nineteen.     How  many  trees  are 
there  in  the  orchard  ? 

3.  P'oar  men  bought  a  piece  of  land  ;  the  first  gave  sixty- 
three  dollars  ;  the  second,  seventy-eight ;  the  third,  forty- 
five  ;  and  the  fourth,  twenty-three.  How  much  did  they 
give  for  the  land  1 

4.  In  an  orchard,  19  trees  bear  cherries,  twenty-eight  bear 
peaches,  8  bear  plums,  and  54  bear  apples.  How  many 
trees  are  there  in  the  orchard  1 

5.  How  many  days  are  there  in  a  year,  there  being  in  Ja- 
nuary 31  days ;  in  February  28  ;  in  March  31 ;  in  April  30 ; 
in  May  31 ;  in  Jun^.  30 ;  in  July  31 ;  in  August  31 ;  in  Sep- 
tember 30;  in  October  31;  in  November  30;  in  Decem- 
ber 31  ? 

6.  The  distance  from  Portland  (in  Maine)  to  Boston,  is 
114  miles ;  from  Boston  to  Providence,  40  miles ;  from 
Providence  to  New  Haven  122  miles;  from  New  Haven  to 
New  York,  88  miles ;  from  New  York  to  Philadelphia,  95 
miles ;  from  Phikdelphia  to  Baltimore,  102  miles ;  from 
Baltimore  to  Charleston,  S.  C.  716  miles ;  from  Charleston 
to  Savannah,  110  miles.  How  many  miles  is  it  from  Port- 
land to  Savannah  1 

7.  What  number  of  dollars  are  there  in  four  bags  ;  the 
first  containing  275  dollars ;  the  second,  356  ;  the  third, 
178  ;  the  fourth,  69  1 

S.  How  many  times  does  the  hammer  of  a  clock  strike  in 
24  hours  ? 

Note.  At  1  o'clock  it  strikes  once,  at  2  o'clock  it  strikes 
twice,  &-C. 

9.  A  man  has  four  horses  ;  the  first  is  worth  sixty-seven 
dollars ;  the  second  is  worth  eighty-four  dollars  ;  the  third  is 
worth  one  hundred  and  twenty  dollars  ;  and  the  fourth  is 
worth  one  hundred  and  eighty-seven  dollars;  and  he  has 
four  saddles  worth  twelve  dollars  apiece.  How  much  are  the 
horses  and  saddles  worth  ? 

10.' A  man  owns  five  houses  ;  for  the  first  he  receives  a 
rent  of  427  dollars  ;  for  the  second,  763  dollars ;  for  the 
third,  654  dollars;  for  the  fourth,  500  dollars ;  and  for  the 
fifth,  325  dollars ;  and  the  rest  of  his  income  is  3,250  dol- 
lars.    What  is  his  whole  income  ? 

11,  A  gentleman  owns  five  farms;  the  first  is  worth 
11,500  dollars;  the  second,  3,057  dollars;  the  third,  2,468 
dollars  ;  the  fourth,  9,462  dollars ;  and  the  fifth,  850  dollars ; 


11.  ADDITION.  15 

and  he  owns  a  house  worth  15,000  dollars,  a  carriacre  worth 
753  dollars,  and  two  horses  worth  175  dollars  apiece.  How 
much  are  they  all  worth  ? 

12  A  merchant  bought  four  pieces  of  cloth,  each  piece 
containing  57  yards.  For  the  first  piece  he  gave  235  dol- 
lars ;  for  the  second,  3S4  dollars  ;  for  the  third,  327  dollars ; 
and  for  the  fourth,  480  dollars.  How  many  yards  of  cloth 
did  he  buy  ?     How  much  did  he  give  for  the  whole  ? 

13.  In  1818  the  navy  of  the  United  States  consisted  of 
three  74s ;  five  44  gun  frigates  ;  three  36s  ;  two  32s ;  one 
20  ;  ten  18s.     How  many  guns  did  they  all  carry  ? 

14.  Suppose  it  requires  65i)  men  to  man  a  74  ;  475  to 
man  a  44  ;  275  to  n.an  a  3G  ;  350  to  man  a  32  ;  200  to  man 
a  20;  and  180  to  man  an  18.  How  many  men  would  it 
take  to  man  the  whole  1 

15.  The  hind  quarters  of  a  cow  weighed  one  hundred  and 
fi<ve  pounds  each  ;  the  fore  quarters  weighed  ninety-four 
pounds  each  ;  the  hide  weighed  sixty-three  pounds  ;  and  the 
tallow  seventy-six  pounds.  What  was  the  whole  weight  of 
the  cow  ? 

16.  A  man  bought  a  barrel  of  flour  for  6  dollars,  and  sold 
it  so  as  to  gain  2  dollars.     How  much  did  he  sell  \l  for  ? 

17.  I  bought  a  (|aantity  of  salt  for  18  dollars,  and  sold  it 
for  7  dollars  more  than  I  gave  for  it ;  how  much  did  I  sell 
it  for  ? 

18.  A  man  bought  three  hogsheads  of  molasses  for  132 
dollars,  and  sold  it  so  as  to  gain  25  dollars  ;  how  much  did 
he  sell  it  for  ? 

19.  A  man  being  asked  his  age,  answered  that  he 
was  twenty-seven  years  old  when  he  was  married,  and 
that  he  had  been  married  thirty-nine  years.  How  old  was 
he? 

20.  A  man  being  asked  his  age,  answered  that  he  had 
passed  the  19  first  years  of  his  life  in  America,  and  that  lie 
had  afterwards  spent  7  years  in  Germany,  13  years  in  France, 
3  years  in  Holland,  and  24  years  in  England.  How  old 
was  he  1 

21.  A  merchant  bought  four  hogsheads  of  wine  for  four 
hundred  and  thirty-seven  dollars,  and  sold  it  again  for  ninety- 
four  dollars  more  than  he  gave  for  it.  How  much  did  he 
sell  it  for  ? 

22.  A  man  commenced  trade  with  three  thousand,  two 
hundred  and  fifty  dollars  ;  after  tradin<T  for  some  time,  he 


16  ARITHMEl'IC.  Part  L 

found  he  had  gained  two  hundred  and  thirty-seven  dollars. 
How  much  had  he  then  ? 

23.  Money  was  first  made  at  Argos,  eight  hundred  and 
ninety-four  years  before  Christ.  How  long  has  it  been  in 
use  at  this  date,  1822  ? 

24.  The  war  between  Great  Britain  and  the  American 
colonies  commenced  in  1775  and  continued  8  years.  In 
what  year  was  the  war  concluded  ? 

25.  General  Washington  was  born  in  the  year  1732,  and 
was  6  years  old  when  he  died.     In  what  year  did  he  die  1 

26.  The  first  tragedy  was  acted  at  Athens,  on  a  cart,'  by 
Thespis,  ^ivq  hundred  and  thirty-six  years  before  Christ. 
How  many  years  is  it  since  ? 

27.  What  was  the  number  of  inhabitants  in  the  New 
England  States,  in  1820,  there  being  in 

Maine  298,335 

New  Hampshire  244,161 

Vermont  235,764 

Massachusetts  523,287 

Rhode  Island  83,059 

Connecticut  275,248  ? 

28.  What  was  the  number  of  inhabitants  in  the  Mid 
die  States,  there  being  in 

New  York  1,372,812 

New  Jersey  277,575 

Pennsylvania  1,049,398 

Delaware  •  72,749 

Maryland  407,350  ? 

29.  What  was  the  number  of  inhabitants  in  the  following 
States,  there  being  in 

Virginia  1,065,366 

North  Carolina  638,829 

South  Carolina  490,309 

Georgia  340,989 

Kentucky  564,317 

Tennessee  422,813 

Alabama  127,901 

Mississippi  75,448 

Louisiana  153.407  ? 

30.  What  was  the  number  of  inhabitants  in  the  following 
States,  there  being  in 

Ohio  581,434 

Indiana  147,178 


III.  MULTIPLICATION.  17 

Illinois  55,211 

Missouri  66,586 

Arkansas  Territory  14,273 

Michigan  Territory  8,896 

District  of  Columbia  33,039  ? 

31.  What  was  the  whole  number  of  inhabitants  in  the 
Uniterl  States  in  1820? 

32.  Add  together  the  following  numbers  ;  32,753  ;  2,047 ; 
840,397  ;  47,640. 

33.  What  is  the  sum  of  the  following  numbers  ;  30  ;  843; 
30,804;  387,6-13;  13;  8,406,127;  4;  900,600? 

34.  What  is  the  sum  of  the  following  numbers,  three  mil- 
lions and  seven  thousand ;  thirty-five  ;  four  hundred  and 
eighty-seven  ;  two  thousand  and  forty-three  ;  ninety-six  mil- 
lions, thirty-four  thousand,  and  forty-two  ;  and  seventeen  1 


MULTIPLICATION. 

III.  1  *  What  will  two  barrels  of  rum  cost,  at  27  dollars 
a  barrel  ? 

2.  What  will  3  hogsheads  of  molasses  amount  to,  at  26 
dollars  a  hogshead  ? 

3.  What  will  14  pounds  of  veal  come  to,  at  4  cents  a 
pound  ? 

4.  What  will  seventeen  pounds  of  beef  cost,  at  five  cents 
a  pound  ? 

5.  What  will  five  cows  cost  r.t  19  dollars  apiece  1 

6.  What  will  3  oxen  cost  at  47  dollars  apiece  ? 

7.  What  cost  15  yards  of  cloth  at  8  dollars  a  yard  ? 

8.  What  cost  26  barrels  of  cider  at  4  dollars  a  barrel  1 

9.  What  cost  98  barrels  of  flour  at  7  dollars  a  barrel  ? 

10.  What  cost  794  barrels  of  flour  at  9  dollars  a  barrel  ? 

11.  There  is  an  orchard  consisting  of  9  rows  of  trees,  and 
there  are  57  trees  in  each  row.  How  many  trees  are  there 
in  the  orchard. 

12.  A  man  bought  8  pieces  of  cloth,  each  piece  contain- 
ing 38  yards,  at  7  dollars  a  yard.  How  many  yards  were 
there,  and  what  did  he  give  for  the  whole  ? 

13.  A  man  bought  9  pieces  of  broadcloth,  each  piece  con- 

•  See  First  Lessons,  sect.  II. 

2  ♦ 


WINE     MEASURE. 

ake         1  pint        marked 

pt. 

1  quart 

1  gallon 

1   barrel  or  half  hhd. 

qt. 

gal. 

bbl. 

1  hogsliead 

hhd. 

1  pipe  or  butt 

p.  or  b. 

1  tun 

T. 

18  ARITHMETIC.  Part  1. 

taining  47  yards,  at  6  dollars  a  yard  ;  and  25  barrels  of  flour 
at  7  dollars  a  barrel.     What  did  he  give  for  the  whole  ? 

14.  A  merchant  bought  a  hogshead  of  wine,  at  the  rate  of 
2  dollars  a  gallon  ;  what  did  it  co.ne  to  ? 


4  gills  (gl.) 

2  pints 

4  quarts 

3H  gallons 

63  gallons 

2  hogsheads 

2  pipes 

By  this  measure  brandy,  spirits,  perry,  cider,  mead,  vine- 
gar, and  oil  are  measured. 

1.5.  At  3  dollars  a  gallon,  what  will  2  pipes  of  wine  cost  ? 

16.  At  4  cents  a  gHl,  what  will  1  pint  of  brandy  cost  1 

17.  At  5  cents  a  gill  what  will   1   quart  of  wane  cost  1 
What  will  1  gallon  cost  1 

Note.     Since  100  cents  make  1  dollar,  it  will  be  easy  to 
tell  how  many  dollars  there  are  in  any  number  of  cents. 

18.  At  8   cents   a  quart,  what  will  1  hhd.   of  molasses 
come  to  1 

19.  How  many  pints  are  there  in  87  quarts  1 

20.  How  many  gills  are  there  in  174  piats  ? 

2i.  How  many  quarts  are  there  in  1  hhd.  of  wine  1 

22.  How  many  quarts  are  there  in  4  hhds.  of  brandy  ? 

2*^.  How  many  pints  are  there  in  one  hhd.  of  molasses  1 

24.  How  many  pints  are  there  in  I  pipe  ? 

25.  How  many  gills  are  there  in  1  hhd  ? 

26.  How  many  gills  are  there  in  1  T.  ? 

27.  How  many  quarts  in  8  gals.  2  quarts  7 

28.  How  many  pints  in  4  gals.  3  qts.  1  pint  X 

29.  How  many  gallons  in  3  hhds.  42  gals.  ? 

30.  How  many  quarts  in  1  p.  1  hhd.  1 

31.  How  many  pints  in  1  hhd.  35  gals.  3  qts.  1  pt.  t 

32    How  many  gills  in  3  hhds.  27  gals.  1  qt.  1  pt.  3  gls.  1 

33.  A  man  having  1  T.  of  wine,  retailed  it  at  5  cents  a 
gill,  how  much  did  it  come  to  ? 

34.  A  man  bought  a  quarter  of  beef,  weighing  237  pounds, 
at  7  cents  a  pound.     How  much  did  it  cost  I 

35.  How  many  are  3  times  784  1 


IV.  MULTIPLICATION.  19 

J?6.  How  many  are  5  times  1,328? 

37.  How  many  are  nine  limes  87,4361 

38.  Multiply  2,487  by  8. 

39.  Multiply  820,438  by  7. 

40.  Multiply  13,052,068  by  5. 

IV.    1.  What  will  18  oxen  cost  at  57  dollars  apiece  1 
Note.     Find  first  what  6  oxen  will  cost,  and  18  oxen  will 

cost  3  times  as  much.     Perform  the  following  examples  in  a 

similar  manner. 

2.  What  would  14  chests  of  tea  cost,  at  87  dollars  a  chest  1 

3.  A  merchant  bought  84  pieces  of  linen,  at  16  dollars 
apiece  ;  how  much  did  it  come  to  1 

4.  A  merchant  bought  15  hogsheads  of  wine,  at  97  dol- 
lars a  hogshead.     How  much  did  the  whole  timount  to  ? 

5.  A  merchant  sold  20  hhds.  of  brandy  at  2  dollars  a  gal- 
lon. How  much  did  each  hogshead  amount  to  ?  How 
much  did  the  whole  amount  to  1 

6.  What  would  28  bales  of  cotton  come  to,  at  75  dollars  a 
bale? 


TIME. 

60  seconds  (sec.)  make         1  minute, 

marked 

min. 

60  minutes                              1  hour 

h. 

24  hours                                  1  day 

d. 

7  days                                    1  week 

w. 

4  weeks                                1  month 

mo. 

13  months  1  day  &- 6  hours,  )  ^ 
or  365  days  and  6  hours        3     ^ 

y- 

12  calendar  months               1  year 

7.  If  a  man  can  earn  eight  dollars  in  a  week,  how  much 
can  he  earn  in  a  year  ? 

8.  If  the  expenses  of  a  man's  family  are  32  dollars  a  week, 
what  will  they  amount  to  in  a  year  ?     What  in  2  years  ? 

9.  How  many  hounj  are  there  in  a  week  ? 

10.  How  many-  minutes  are  there  in  a  day  ? 

11.  How  many  minutes  are  there  in  a  week  ? 

12.  How  many  hours  are  there  in  2  mo.  3d.? 

13.  If  a  man  can  travel  7  miles  in  an  hour,  how  far  caa 
he  travel  in  8  days,  when  the  days  are  9  hours  long  ? 

14.  If  a  ship  sail  11  miles  in  an  hour,  how  far  would  it 
sail  at  that  rate  in  one  day,  or  24  hours  ? 

15.  If  a  ship  sail  8  miles  in  an  hour,  how  far  would  it 
sail  at  that  rate  in  18  days  ? 


20 


ARfTHMETIC. 


Parti 


16.  Suppose  a  cistern  has  a  cock  which  conveys  37  gal- 
lons into  it  in  an  half  hour,  how  much  would  run  into  it  in  1 
d.  8  h. 

17.  If  a  man  can  earn  18  dollars  in  a  calendar  month,  how 
much  would  he  earn  in  7  y.  8  mo.  ? 

18.  In  one  year  how  many  minutes  ? 

19.  In  two  y.  3  mo.  18  d.  how  many  days  1 

20.  A  cannon  ball  at  its  first  discharge,  flies  at  the  rate 
of  about  9  miles  in  a  minute ;  how  far  would  it  fly  at  that 
rate  in  24  hours  1     How  far  in  15  days  ? 


21. 

Multiply        67  by  14|32.     M-iltipIy    21,378  by  36 

22. 

321 

1533. 

825      42 

23. 

463 

16 

34. 

164      45 

24. 

275 

18 

35. 

1,163      48 

25. 

144 

21 

36. 

9,876      49 

26. 

2,107 

24 

37. 

40,073      54 

27. 

381 

25 

38. 

3,502      m 

28. 

1,234 

27 

39. 

127      63 

29. 

3,002 

28 

40. 

308      72 

30. 

4,381 

32 

41. 

1,437      81 

31. 

11,962 

35 

42.  What  would  17  loads  of  hay  come  to  at  26  dollars  a 
load? 

Note.  First  find  the  price  of  16  loads,  and  then  add  the 
price  of  1  load.  Perform  the  following  examples  in  a  similar 
manner. 

43.  What  would  17  oxen  cost,  at  87  dollars  apiece  1 

44.  What  would  87  pounds  of  tobacco  cost,  at  23  cents  a 
pound  ? 

45.  What  would  28  pounds  of  sugar  cost,  at  13  cents  a 
pound  1 

46.  What  would  59  bushels  of  potatoes  cost,  at  38  cents  a 
bushel  ? 

47.  What  costs  1  hhd.  of  molasses  at  37  cents  a  gallon  1 


48.        Mu 

Itiply        19  by  17 

52. 

Multiply 

206  by  38 

49. 

37       19 

53. 

314      47 

50. 

106      23 

54. 

203      58 

51. 

141       34 

55. 

715      67 

V.     1.  What  cost  5  pounds  of  beef  at  10  cents  a  pound  ? 
2.  What  will  12  barrels  of  flour  come  to,  at  10  dollars  a 
barrel  1 


V.  MULTIPLICATION.  21 

Note.  Observe  that  when  you  multiply  by  10,  it  is  done 
by  annexing  a  zero  to  the  right  of  the  number  ;  and  by  100, 
it  is  done  by  annexing  two  zeros,  &>c. ;  and  find  the  reason 
why. 

3.  What  would  a  hogshead  of  wine  come  to,  at  ten  cents 
a  pint  ? 

4.  If  10  men  can  do  a  piece  of  work  in  7  days,  how  many 
days  will  it  take  1  man  to  do  it  ? 

5.  What  would  an  ox,  weighing  873  pounds,  come  to,  at 
10  cents  a  pound  ? 

6.  If  100  men  were  to  receive  8  dollars  apiece,  how  many 
dollars  would  they  all  receive  1 

7.  If  27  men  were  to  receive  100  dollars  apiece,  how  many 
dollars  would  they  all  receive  1 

FEDERAL  MONEY. 

10  mills  (m.)  make    1  cent    marked  c. 

10  cents  1  dime  d. 

10  dimes  1  dollar  dol.  or  $. 

10  dollars  1  eagle  E. 

8.  In  3  dimes  how  many  cents  ? 

9.  In  5  dollars  how  many  dimes  ?     How  many  cents  I 

10.  In  17  dollars  how  many  cents  1 

11.  In  83  cents  how  many  mills  ? 

12.  In  753  dols.  how  many  cents  ? 

13.  In  1  dol.  how  many  mills  1 

14.  In  84  dols.  how  many  mills  1 

15.  In  7  dols.  and  53  cents,  how  many  cents? 

16.  In  183  dols.  and  14  cents,  how  many  cents  ? 

17.  In  283  dols.  43  cents  and  8  mills,  how  many  mills? 

18.  In  8246  dols.  2  d.  5  c.  6  m.  how  many  mills  1 

It  is  usual  to  write  dollars  and  cents  in  the  following  man- 
ner :  43  dols.  5.  d.  7  c.  and  4  mills,  is  written  843.574. 
The  character  $  written  before  shows  that  it  is  federal  mo- 
ney. The  figures  at  the  left  of  the  point  (.)  are  so  many 
dollars,  the  Prst  figure  at  the  right  of  the  point  is  so  many 
dimes,  the  next  so  many  cents,  and  the  third  so  many  mills. 

It  may  be  observed  that  when  dollars  stand  alone,  they 
are  changed  to  dimes  by  annexing  one  zero  to  the  right,  be- 
cause that  multiplies  them  by  10.  They  are  changed  to 
cents  by  annexing  two  zeros,  because  that  multiplies  theni 


23  ARITHMETIC.  Part  I. 

by  100.  They  are  changed  to  mills  by  annexing  three  ze- 
ros, because  that  multiplies  them  by  1,000.  Thus  43  dol- 
lars are  430  dimes,  4,300  cents,  or  43,000  mills.  5  dimes 
are  50  cents,  or  500  mills.  7  cents  arc  70  mills.  The 
above  example  then  may  be  read  43  dols.  57  cents  and  4 
mills;  or  435  dimes,  7  cents,  and  4  mills;  or  4,357  cents 
and  4  mills ;  or  43,574  mills.  When  there  are  dollars, 
dimes,  and  cents,  the  figures  on  the  left  of  the  point  may  be 
read  dollars,  and  those  on  the  right,  cents ;  or  they  may  be 
all  read  together  as  cents.  When  the  number  of  cents  ex- 
ceeds 100,  they  are  changed  to  dollars  by  putting  a  point 
between  the  second  and  third  figures  from  the  right.  If 
there  are  mills  in  the  number,  all  the  figures  may  be  read 
together  as  mills.  Any  number  of  mills  are  changed  to  dol- 
lars by  putting  a  point  between  the  third  and  fourth  figure 
from  the  right ;  the  figures  at  the  left  will  be  dollars,  and 
those  at  the  right,  dimes,  cents,  and  mills.  Since  any  sum 
which  has  cents  or  mills  in  it,  may  be  considered  as  so  many 
cents  or  mills,  it  is  evident  that  any  operation,  as  addition, 
multiplication,  &c.  may  be  performed  upon  it  in  the  same 
manner  as  upon  sicnple  numbers. 

If  the  sum  consists  of  dollars  and  a  number  of  cents  less 
than  ten,  there  must  be  a  zero  between  the  dollars  and  the 
cents  in  the  place  of  dimes.  Thus  7  dols.  and  5  cents  must 
be  written  $7.05. 

19.  What  will  10  yards  of  cloth  cost  at  $4..53  a  yard  1 

20.  What  will  10  pounds  of  coffee  cost  at  $0.27  a  pound  1 

21.  What  will  100  sheep  cost  at  88.45  apiece  ? 

22.  What  will  1,000  yards  of  cloth  cost  at  $0.35  a  yard  ? 

23.         Multiply 

24. 

25. 

26. 

27. 

28. 

29. 

30. 

31. 

VI.     1.  What  cost  75  lb.   of  tobacco  at  20  cents  a  pound  ? 

2.  What  cost  30  cords  of  wood  at  $0,75  a  cord  ? 

3.  If  400  men  receive  135  dollars  apiece,  how  many  dol- 
lars will  thfy  all  receive  1 


5  by  10 

32. 

Multiply  90 

by    100 

47   10 

33. 

4 

1,000 

30   10 

34. 

73 

1,000 

124   10 

35. 

80 

1,000 

387   10 

36. 

132 

1,000 

450   10 

37. 

800 

1,000 

13,008   10 

38. 

1,643 

1,000 

7  100 

39. 

725 

10,000 

38  100 

40. 

76,438 

10,000 

VII.  MULTIPLICATION.  23 

4.  If  30  men  can  do  a  piece  of  work  in  43  days,  liow 
many  days  will  it  take  1  man  to  do  it  ? 

5.  If  70  men  can  do  a  piece  of  work  in  83  days,  how 
many  men  will  it  take  to  do  it  in  one  day  1 

6.  If  the  pendulum  of  a  clock  swing  once  in  a  second^ 
how  many  times  will  it  swing  in  an  hour  1  How  many 
times  in  a  day  ?     How  many  times  in  a  week  ? 

7.  How  many  seconds  are  there  in  10  min.  23  sec.  ? 

8.  How  many  minutes  are  there  in  7  h.  23  min.  ? 

9.  How  many  minutes  are  there  in  3d.  7  h.  43  min.  1 

10.  Plow  many  seconds  are  there  in  8  d.  7  h.  34  min. 
19  sec.  ? 

11.  A  garrison  of  3,000  men  are  to  be  paid,  and  each 
man  is  to  receive  128  dollars.  How  many  dollars  will  they 
all  receive  ? 

12.  What  cost  30  barrels  of  cider  at  $3.50  a  barrel  ? 

13.  There  are  320  rods  in  a  mile,  how  many  rods  are 
there  in  7  miles  1  How  many  in  10  miles  1  How  many  in 
30  miles  1     How  many  in  500  miles  ? 


14.  Multiply    34  by        20 

15.  57  300 

16.  250  60 
\7.  387  5,000 


18.  Multiply  4,007  by      80 

19.  11,600  700 

20.  4,960  40,000 

21.  13,400  8,000 


VII.     1.  What  will  17  oxen  come  to  at  42  dollars  apiece  1 
Note.     Find  the  price  of  10  oxen  and  of  7  oxen  sepa- 
rately, and  then  add  them  together. 

2.  What  will  34  barrels  of  flour  come  to,  at  $6.43  a 
barrel  ? 

Note.  Find  the  price  of  30  barrels  and  of  4  barrels  sepa- 
rately, and  then  add  them  together. 

3.  What  cost  19  gallons  of  wine,  at  $1.28  a  gallon  1 

4.  What  cos-t  68  yards  of  cloth,  at  $9.36  a  yard  ? 

5.  What  will  87  thousand  of  boards  come  to,  at  $5.50  a 
thousand  ? 

6.  What  will  58  barrels  of  beef  come  to,  at  $9.75  a  barrel  ? 

7.  What  will  87  gallons  of  brandy  come  to,  at  $1.60  a 
gallon  1 

8.  A  and  B  depart  from  the  same  place  and  travel  in  op- 
posite directions,  A  at  the  rate  of  38  miles  in  a  day,  and  B 
at  the  rate  of  42  miles  a  day.  How  far  apart  will  they  be  at 
the  end  of  the  first  day  1     How  far  at  the  end  of  15  days  1 


n  ARITHMETIC.  Part  1. 

9.  What  will  287  barrels  of  turpentine  come  to,  at  $3.25 
a  barrel  1 

Note.  Find  the  price  of  200  barrels,  of  SO  barrels,  and  of 
7  barrels  separately,  and  then  add  them  together. 

10.  What  will  358  barrels  of  beef  come  to,  at  $7.55  a 
barrel  ? 

11.  A  drover  bought  853  sheep  at  an  average  price  ol 
$3.58  apiece.     What  were  the  whole  worth  ? 

12.  A  merchant  bought  105  hundred  weight  of  lead,  at 
$17.33  a  hundred  weight ;  how  much  did  the  whole  come  to  1 

13.  If  a  ship  sail  8  miles  in  an  hour,  how  many  miles 
will  she  sail  in  a  day,  at  that  rate  ?     How  far  in  127  days  1 

14.  An  army  of  8,975  men  are  to  receive  138  dollars 
apiece.     How  many  dollars  will  they  all  receive  ? 

15.  An  army  of  11,327  men  are  to  receive  a  year's  pay, 
at  the  rate  of  5  dollars  a  month  for  each  man.  How  many- 
dollars  will  they  all  receive  1 

16.  Bought  207  chaldrons  of  coal,  at  $12,375  a  chaldron. 
How  much  did  it  come  to  1 

17.  Bought  857  pounds  of  sugar  at  $0,125  a  pound. 
How  much  did  it  come  to  1 

18.  Shipped  350  casks  of  butter  worth  $14.50  a  cask. 
What  was  the  value  of  the  whole  ? 

19.  What  cost  354  fother  of  lead,  at  $63.57  a  fother  ? 

20.  What  cost  25,837  gallons  of  brandy,  at  $2,375  a 
gallon  ? 

21.  If  it  cost  $28.56  to  clothe  a  soldier  1  year,  how  many 
dollars  will  it  cost  to  clothe  an  array  of  15^200  men  the  same 
time? 

by  47 

250 

308 

1,005 

2,700 

38,400 

30,704 

37,000 

300,005 

703,004 

Miscellaneous  Examples. 
1.  If  1  pound  of  tobacco  cost  28  cents,  what  will  a  keg 
of  tobacco,  weighing  112  pounds,  cost  1 


22. 

Multiply            887 

23. 

6,300 

24. 

1,006 

25. 

15,030 

26. 

38,446 

27. 

487,500 

28. 

7,035,064 

29. 

9,800,000 

30. 

78,508,060 

31. 

43,060,085 

VII.  MULTIPLICATION.  25 

AVOIRDUPOIS    WEIGHT. 

"^-.    16  drams    (dr.)     make  1  ounce,      marlsed  oz. 

16  ounces  1  pound  lb. 

28  pounds  1  quarter  qr. 

4  quarters  1  hundred  weight  cwt. 

20  hundred  weight  1  ton  T. 

By  this  weight  are  weighed  all  things  of  a  coarse  and 
drossy  nature ;  such  as  butter,  cheese,  flesh,  grocery  wares, 
and  all  metals  except  gold  and  silver. 

2.  At  12  cents  per  lb.  how  much  will  1  quarter  of  sugar 
come  to  ? 

3.  If  1  quarter  of  sugar  cost  7  dollars,  what  will  1  cwt. 
cost? 

4.  How  many  pounds  are  there  in  1  cwt.  1 

5.  In  2  cwt.  2  qrs.  how  many  quarters  ? 

6.  In  3  qrs.  18  lb.    how  many  pounds  1 

7.  In  2  cwt.  1  qr.  how  many  pounds  1 

8.  In  1  cwt.  3  qrs.  23  lb.    how  many  l^ounds  ? 

9.  In  18  lb.    how  many  ounces  1 

10.  In  12  cwt.  how  many  ounces  ? 

11.  In  14  cwt.  3  qrs.  15  lb.    8  oz.  how  many  ounces  1 

12.  At  9  cents  a  pound,  what  cost  3  cwt.  2  qrs.  16  lb*  of 
sugar  1 

TROY    WEIGHT. 

24  grains   (gr.)   make    1  penny-weight,   marked   dwt. 
20  penny-weights  1  ounce  oz. 

12  ounces  1  pound  lb. 

By  this  weight  are  weighed  gold,  silver,  jewels,  corn, 
bread,  and  liquors. 

13.  If  an  ingot  of  silver  weigh  42  oz.  13  dwt.,  what  is  it 
worth  at  4  cents  per  dwt.  1 

14.  What  is  the  value  of  a  silver  cup  weighing  9  oz.  4 
dwt.  16  gr.  at  3  mills  per  grain  1 

15.  In  15  ingots  of  gold  each  weighing  9  oz.  5  dwt.  7 
gr.  how  many  grains  1 

apothecaries'    WEIGHT. 

20  grains    (gr.)    make  1  scruple,  marked    sc. 

3  scruples  1  drpm  dr.  or  5 

8  drams  1  ounce  oz.  or  § 

12  ounces  1  pound  lb. 
3 


26  ARITHMETIC.  Part  1. 

Apothecaries  use  this  weight  in  compounding  their  medi- 
cines, but  they  buy  and  sell  by  Avoirdupois  weight  Apo- 
thecaries' is  the  same  as  Troy  weight,  having  only  some  dif- 
ferent divisions. 

16.  In  9 lb.  8   §.1  5.  2  sc.  19  gr.  how  many  grains  1 


DRY    MEASURE. 

2  pints    (pt.)  make      I  quart,    marked  qt. 

8  quarts  1  peck  pk. 

4  pecks  1  bushel  bu. 

8  bushsis  1  quarter  qr. 

By  this  measure,  salt,  ore,  oysters,  corn,  and  other  dry 
goods  are  measured. 

17.  At  43  cents  a  peck,  what  cost  14  bu.  3  pks.  of  wheat? 

18.  At  3  cents  a  quart  what  will  5  bu.  2  pks.  3  qts.  of  salt 
come  to  1 

CLOTH    MEASURE. 


2j  inches  (in.)   mal 

ke      1  nail,     marked      nl. 

4    nails 

1  quarter                  qr. 

4    quarters 

1  yard                       yd. 

3    quarters 

.    1  ell  Flemish           Ell  Fl. 

5    quarters 

1  ell  English            Ell  Eng. 

5    quarters 

1  aune  or  ell  French. 

19.  At  '27  cents  a  nail,  what  is  the  price  of  2  yds.  1  qr.  3 
nls.  of  cloth  . 

20.  If  1  qr.  cost  |2,50,  what  cost  43  ells  English  of  broad- 
cloth ? 

21.  At  42  cents  a  nail,  what  cost  13  ells  Fl.  3  qrs.  of 
broadclotk  \ 

22.  How  many  seconds  are  there  in  4  years  1 

23.  How  man)  seconds  are  there  in  8  y.  3  mo.  2  wks.  2 
d.  19  h.  43  min.  57  sec.  ? 

24.  How  many  calendar  months  are  there  from  the  1st 
Feb.  1819,  to  the  1st  August,  1822  1 

25.  How  many  days  are  there  from  the  7th  Sept.  1817,  to 
the  17th  May,  1822? 

26.  How  many  minutes  are  there  from  the  13th  July,  at 
43  minutes  after  9  in  the  morning,  to  the  5th  Nov.  at  19 
min.  past  3  in  the  afternoon  ? 


Vir.  MULTIPLICATION.  27 

27.  How  many  seconds  old  are  you  ? 

28.  How  many  seconds  from  the  commencement  of  the 
Christian  era  to  the  year  1822  ? 

29.  At  4  cents  an  ounce,  how  much  would  3  cwt.  2  qrs. 
18  lb.  7  oz.  of  snuff  come  to? 

30.  At  28  cents  a  pound,  what  would  3  T.  2  cwt.  3  qrs. 
16  lb.  of  tobacco  come  to  ? 

31.  If  a  cannon  ball  flies  8  miles  in  a  minute,  how  far 
would  it  fly  at  that  rate  in  7  y.  2  mo.  3  wks.  2  days  1 

32.  If  a  quantity  of  provision  will  last  324  men  7  days, 
how  many  men  will  it  last  one  day  1 

33.  A  garrison  of  527  men  have  provision  sufficient  to  last 
47  days,  if  each  man  is  allowed  15  oz.  a  day  ;  how  many 
days  would  it  last  if  each  man  were  allowed  only  1  oz.  a 
day? 

34.  A  garrison  of  527  men  have  provision  sufficient  to  last 
47  days,  if  each  man  is  allowed  15  oz.  a  day ;  how  many 
men  would  it  serve  the  same  time,  if  each  man  were  allow- 
ed only  1  oz.  a  day  ? 

35.  If  a  man  performs  a  journey  in  58  days,  by  travelling 
9  hours  in  a  day,  how  manv  hours  is  ho  performing  it  ? 

36.  If  by  working  13  hours  in  a  day  a  man  can  perform 
a  piece  of  work  in  217  days  ;  how  long  would  it  take  him  to 
do  it  if  he  worked  only  I  hour  in  a  day  ? 

37.  If  by  labouring  14  hours  in  a  day  237  men  can  build 
a  ship  in  132  days,  how  many  days  would  it  take  them,  if 
they  work  only  1  hour  in  a  day  ?  How  many  men  would 
it  take  to  do  it  in  132  days,  if  they  work  only  1  hour  in  a 
day  ? 

38.  How  many  yards  of  cloth  that  is  1  qr.  wide,  are  equal 
to  27  yaids  that  is  1  yd.  wide  ? 

39.  If  a  piece  of  cloth  that  is  1  qr.  wide  is  worth  $67.25, 
what  is  a  piece  containing  the  same  number  of  yards  of  the 
same  kind  of  cloth  worth,  that  is  1  yd.  wide  ? 

40.  If  a  bushel  of  wheat  afford  65  eight-penny  loaves, 
how  many  penny  loaves  may  be  obtained  from  it  ? 

41.  What  is  the  price  of  4  pieces  of  cloth,  the  first  con- 
taining 21  yards,  at  $4.75  a  yard  ;  the  second  containing  27 
yards,  at  $7.25  a  yard ;  the  third  containing  IS  yards,  at 
$9.00  a  yard  ;  and  the  fourth  containing  32  yards,  at  $8.57 
a  yard  ? 

42.  A  man  bought  1.5  lb.  of  beef,  at  9  cents  a  pound ; 
28  lb,    of  sugar,  at  $0,125  a  pound;  18  gallons  of  wine,  at 


28  ARITHMETIC.  Part  1. 

$1.5G  a  gallon  ;  a  barrel  of  flour,  for  $8.00  ;  and  3  barrels 
of  cider,  at  $3.50  a  barrel.  How  much  did  the  whole 
amount  to  ? 

Interest  is  a  reward  allowed  by  a  debtor  to  a  creditor  for 
the  use  of  money.  It  is  reckoned  by  the  hundred,  hence 
the  rate  is  called  so  much  per  cent,  or  per  centum.  Per 
centum  is  Latin,  signifying  by  the  hundred.  6  per  cent, 
signifies  6  dollars  on  a  hundred  dollars,  6  cents  on  a  hun- 
dred cents,  £G  o\^  =£100,  &lz.  so  5  per  cent,  signifies  5  dol- 
lars on  100  dollars,  &l(i.  Insurance j  commission,  and  pre- 
miums of  every  kind  are  reckoned  in  this  way.  Discount  is 
so  much  per  cent,  to  be  taken  out  of  the  principal. 

43.  If  1  dollar  gain  6  cents  interest  a  year,  how  much  will 
13  dollars  gain  in  the  same  time  1 

44.  What  is  the  interest  of  $43.00  for  1  year  at  6  per 
cent.  1 

45.  What  is  the  interest  of  $157.00  for  1  year  at  5  per 
cent.  1 

46.  What  is  the  interest  of  $1.00  for  2  year,s  at  6  per 
cent.  ?     What  for  5  years  ? 

47.  What  is  the  interest  of  $247.00  for  3  years  at  7  per 
cent  1 

48.  How  much  must  I  give  for  insuring  a  ship  and  cargo 
worth  $150,000.00  at  2  per  cent.  ? 

49.  Imported  some  books  from  England,  for  which  I 
paid  $150.00  there.  The  duties  in  Boston  were  15  per 
cent.,  the  freight  $5.00.     What  did  the  books  cost  me  ? 

50.  What  must  I  receive  for  a  note  of  $275.00  that  has 
been  due  3  years,  interest  at  6  per  cent.  1 

51.  A  man  failing  in  trade,  is  able  to  pay  only  $0.68  on  a 
dollar  ;  how  much  can  he  pay  on  a  debt  of  $5  dollars  1 
How  much  on  a  debt  of  20  dollars  1 

52.  A  man  failing  in  trade,  is  able  to  pay  only  $0.73  on  a 
dollar  ;  how  much  will  he  pay  on  a  debt  of  $47.00  1  How 
much  on  a  debt  of  $123.00?  How  much  on  a  debt  of 
$2,500.00  ? 

53.  A  merchant  bought  a  quantity  of  goods  for  243  dol- 
lars, and  sold  them  so  as  to  gain  15  per  cent. ;  how  much 
did  he  gain,  and  how  much  did  he  sell  them  for  1 

54.  A  merchant  bought  a  quantity  of  goods  for  $843.00  % 
liow  much  must  he  sell  them  for  to  gain  23  per  cent,  1 


VIII.  SUBTRACTION.  39 


SUBTRACTION. 

VIII.  1.*  David  had  nine  peaches,  and  gave  four  of 
them  to  George  ;  how  many  had  he  left  1 

2.  A  man  having  15  dollars,  lost  9  of  them  ;  how  many 
had  he  left  ? 

3.  David  and  William  counted  their  apples ;  David  had 
35,  and  William  had  17  less ;  how  many  had  William  1 

4.  A  man  owing  48  dollars,  paid  29  ;  how  many  did  he 
then  owe  ? 

5.  A  man  owing  48  dollars,  paid  all  but  19  ;  how  many 
did  he  pay  ? 

6.  A  man  owing  a  sum  of  money,  paid  29  dollars,  and 
then  he  owed  19  ;  how  many  did  he  owe  at  first  ? 

7.  A  man  being  asked  how  old  he  was  when  he  was  mar- 
ried, answered,  that  his  present  age  was  sixty-four  years, 
and  that  he  had  been  married  37  years ;  what  was  his  age 
when  he  was  married  ? 

8.  A  man  being  asked  how  long  he  had  been  married, 
answered,  that  his  present  age  was  sixty-four  years,  and  that 
he  was  twenty-seven  years  old  when  he  was  married  ;  hov/ 
long  had  he  been  married  ? 

9.  A  man  being  asked  his  age,  answered,  that  he  was  27 
years  old  when  he  was  married,  and  that  he  had  been  mar- 
ried 37  years.     What  was  his  age  ? 

10.  A  man  bought  a  piece  of  cloth  containing  93  yards, 
and  sold  45  yards  of  it ;  how  many  yards  had  he  left  ? 

11.  A  merchant  bought  a  piece  of  cloth  for  one  hundred 
and  fifteen  dollars,  and  sold  it  again  for  one  hundred  and 
thirty-eight  dollars.    How  much  did  he  gain  by  the  bargain  1 

12.  A  merchant  sold  a  piece  of  cloth  for  138  dollars, 
which  was  23  dollars  more  than  he  gave  for  it ;  how  much 
did  he  give  for  it  1 

13.  A  merchant  bought  a  piece  of  cloth  for  115  dollars, 
and  sold  it  so  as  to  lose  23  dollars.  How  much  did  he  sell 
It  for  1 

14.  A  man  bought  a  quantity  of  wine  for  753  dollars,  but 
aot  being  so  good  as  he  expected,  he  was  willing  to  lose  87 
dollars  in  the  sale  of  it ;  how  much  did  he  sell  it  for  ? 

15    A  man  owing  two  thousand,  six  hundred,  and  forty- 

*  See  First  Lessons,  sect.  1, 
3* 


30  ARITHMETIC.  Part  1. 

three  dollars,  paid  at  several  times  as  follows ;  at  one  time 
two  hundred  and  seventy-five  dollars  ;  at  another  fifty-eight 
dollars  ;  at  another  seven  dollars  ;  and  at  another  one  thou- 
sand and  sixty-seven  dollars ;  how  much  did  be  then  owe  1 

16.  From  Boston  to  Providence  it  is  41  miles,  and  from 
Boston  to  Attleborough  (which  is  upon,  the  road  from  Bos- 
ton to  Providence)  it  is  28  miles  ;  how  far  is  it  from  Attle- 
borough to  Providence  ? 

17.  From  Boston  to  New  York  it  is  250  miles ;  suppose  a 
man  to  have  set  out  from  Boston  for  New  York,  and  to 
have  travelled  14  hours,  at  the  rate  of  five  miles  in  an  hour ; 
how  much  farther  has  he  to  travel  ? 

18.  General  Washington  was  born  A.  D.  1732,  and  died 
-in  1799  ;  how  old  was  he  when  he  died  ? 

19.  Dr.  Franklin  died  A.  D.  1790,  and  was  84  years  old 
when  he  died  ;  in  what  year  was  he  born  1 

20.  A  gentleman  gave  853  dollars  for  a  carriage  and  two 
horses  ;  the  carriage  alone  was  valued  at  387  dollars  ;  what 
was  the  value  of  the  horses  ?  How  much  more  were  the 
horses  worth  than  the  carriage  ? 

21.  A  man  died  leaving  an  estate  of  eight  thousand,  four 
hundred,  and  twenty-three  dollars  ;  which  he  bequeathed  as 
follows  ;  two  thousand,  three  hundred  dollars  to  each  of  his 
two  daughters,  and  the  rest  to  his  son  ;  what  was  the  son's 
share  ? 

22.  A  gentleman  bought  a  house  for  sixteen  thousand, 
and  twenty-eight  dollars ;  a  carriage  for  three  hundred  and 
eight  dollars,  and  a  span  of  horses  for  five  hundred  and 
eighty-three  dollars.  He  paid  as  follows ;  at  one  time  nine- 
ty-seven dollars ;  at  another,  one  thousand,  and  eight  dol- 
lars ;  and  at  a  third,  four  thousand,  two  hundred,  and  six 
dollars.     How  much  did  he  then  owe  ? 

23.  In  Boston,  by  the  census  of  1820,  there  were  43,278 
inhabitants ;  in  New  Y'ork,  123,706.  How  many  more  in- 
habitants were  there  in  New  Y'ork  than  in  Boston  ? 

24.  In  Boston,  by  the  census  of  1810,  the  number  of  in- 
habitants was  33,250 ;  and  in  1820  it  was  43,278.  AVhat 
was  the  increase  for  10  years  1 

25.  A  merchant  bought  2  pipes  of  brandy  for  042  dollars 
aud  retailed  it  at  3  dollars  a  gallon.     How  much  did  he  gain  1 

2G.  A  man  bought  359  kegs  of  tobacco,  at  9  dollars  a 
keg  ;  654  barrels  of  beef,  at  8  dollars  a  barrel  ;  9  bags  of 
coffee,  at  29  dollars  a  bag      In  exchange  he  gave  3  hhds. 


VIII.  SUBTRACTION.  31 

of  brandy,  at  2  dollars  a  gallon  ;  473  cwt.  of  sugar,  at  8  dol- 
lars per  cwt.     How  much  did  he  then  owe  1 

27.  A  man  bought  7  lb.  of  sugar,  at  ^0.125  per  lb. ;  4 
gals,  of  molasses,  at  0.375  per  gall.  ;  5  lb.  of  raisins,  at 
$0.14  per  lb.  ;  1  bbl.  of  flour,  for  $6.00.  He  paid  a  ten 
dollar  bill ;  how  much  change  ought  he  to  receive  back  ? 

28.  Two  merchants,  A  and  B,  traded  as  follows  ;  A  sold 
B  24  pipes  of  wine,  at  $  1 .87  per  gal. ;  and  B  sold  A  32 
hhds.  of  molasses,  at  $47.00  per  hhd.     The   balance  was 
paid  in  money  ;  how  much  money  was  paid,  and  which  re 
ceived  it  ? 

29.  A  merchant  sold  35  barrels  of  flour,  at  7  dollars  per 
barrel ;  but  for  ready  money  he  made  10  per  cent,  discount. 
How  much  did  the  flour  come  to  after  the  discount  was 
made? 

30.  A  merchant  bought  15  'hhds.  of  wine,  at  $2.00  per 
gallon  ;  but  not  finding  so  ready  a  sale  as  he  wished,  he  was 
obliged  to  sell  it  so  as  to  lose  8  per  cent,  on  the  cost.  How 
much  did  he  lose,  and  how  much  did  he  sell  the  whole  for  1 

31.  Suppose  a  gentleman's  income  is  $1,836.00  a  year, 
and  he  spends  $3.27  a  day,  one  day  with  another;  how 
much  will  he  spend  in  the  year  1  How  much  of  his  income 
will  he  save  1 

32.  What  is  the  difference  between  487,068  and  24,703  ? 

33.  How  much  larger  is  380,064  than  87,065  ? 

34.  How  much  smaller  is  8.756  ihai;  3*',005,078  ? 

35.  How  much  must  you  add  to  7,643  to  make  16,487  ? 

36.  How  much  must  you  subtract  from  2,483  to  leave 
527? 

37.  If  you  divide  3,880  dollars  between  two  men,  giving 
one  1,907  dollars,  how  much  will  you  give  the  other  ? 

38.  Subtract  38,506  from  90.000. 

39.  Subtract  20,07r>  from  180,003. 

40.  A  man  having  1,000  dollars,  gave  away  one  dollar; 
how  many  dollars  had  he  left  ? 

41.  A  man  having  $1,000.00,  lost  seventeen  cents,  how 
much  had  he  left  ? 

42.  Whai  is  the  difference  between  13  and  800,060  ? 

43.  What  is  the  difference  between  160,000  and  70? 

44.  How  much  must  you  add  to  123  to  make  10,000  ? 

45.  A  man's  income  is  $2,738.43  a  year,  and  he  spends 
§1,897.57  ;  how  much  does  he  save  ? 

46.  Subtract  93  from  80,640. 


32  ARITHMETIC.  Part  1. 

47.  A  merchant  shipped  molasses  to  tne  amount  of 
$15,000.00,  but  during  a  storm  the  master  was  obhged  to 
throw  overboard  to  the  amount  of  8853.42 ;  what  was  the 
Talue  of  the  remaining  part  T 

48.  A  man  bought  goods  to  the  amount  of  -$1,153.00,  at 
6  months'  credit,  but  preferring  to  pay  ready  money,  a  dis- 
count was  made  of  $35.47.     What  did  he  pay  for  the  goods  ! 

49.  Subtract  one  cent  from  a  thousand  dollars. 


DIVISION. 

IX.     1.  How  many  oranges,  at  6  cents  apiece,  can  you 
buy  for  36  cents  1 

2.  How  many  barrels  of  cider,  at  3  dollars  a  barrel,  can 
be  bought  for  27  dollars  ? 

3.  How  many  bushels  of  apples,  at  4  shillings  a  bushel, 
can  you  buy  for  56  shillings  ? 

4.  How  many  barrels  of  flour,  at  7  dollars  a  barrel,  can 
you  buy  for  98  dollars  ? 

5.  How  many  dollars  are  there  in  96  shillings  % 

ENGLISH    MONEY. 

4  farthings    (qr.)    make    1  penny,    marked    d. 


12  pence 


1  shilling  s. 


20  shillings  1  pound  £ 

21  shillings  1  guinea. 

This  money  was  used  in  this  country  until  A.  D.  1786, 
when,  by  an  act  of  Congress,  the  present  system,  which  is 
called  Federal  Money,  was  adopted.  Some  of  these  denorni- 
nations,  however,  are  still  used  in  this  country,  as  the  shil- 
ling and  the  penny,  but  they  are  different  in  value  from  the 
English.  In  English  money  4s.  6d.  is  equd  in  value  to  the 
Spanish  and  American  dollar.  But  a  dollar  is  called  six 
shillings  in  New  England ;  eight  shillings  m  New  York  ; 
and  7s!  6d.  in  New  Jersey.  The  English  guinea  is  equal  to 
28s.  in  New  England  currency.  The  dollar  will  be  con- 
sidered 6s.  in  this  book,  unless  notice  is  given  of  a  different 
value. 

6.  How  many  pence  are  there  m  84  farthings  1 


IX.  DIVISION.  33 

7.  How  many  lb.  of  sugar,  at  9d.  per  lb.,  may  be  bought 
for  117d. 

8.  How  much  beef,  at  8  cents  per  lb.,  may  be  bought  for 
$1.12? 

9.  How  many  lb.  of  steel,  at  13  cents  per  lb.,  may  be 
bought  for  82.21  ? 

10.  How  many  cwt.  of  sugar,  at  $14  per  cwt.,  may  be 
bought  for  $280  1 

11.  How  many  cwt.  of  cocoa,  at  $17  per  cwt.,  may  be 
bought  for  $391  ? 

12.  How  much  cocoa,  at  $25  per  cwt.,  may  be  bought 
for  475  dollars  ? 

13.  How  much  sugar,  at  8d.  per  lb.,  may  be  bought  for 
4s.  8d.  ? 

14.  How  much  cloth,  at  4s.  per  yard,  may  be  bought  for 
\£.  12s.  ? 

15.  How  much  snuff,  at  2d.  2  qr.  per  oz.,  may  be  bought 
for  40  farthings  ? 

IG.  How  much  wheat,  at  8s.  per  bushel,  may  be  bought 
for  2^.  16s.  1 

17.  How  much  cloth,  at  7s.  per  yard,  may  be  bought  for 
3^.  17s. 

18.  How  much  pork,  at  9d.  per  pound,  may  be  bought  for 
l£.  4s.  9d.  ? 

19.  How  much  molasses,  at  1  Id.  per  quart,  may  be  bought 
for2ir.  15s.  Ud. 

20.  In  38  shillings  how  many  pounds  1  . 

21.  In  53  shillings  how  many  pounds  1 

22.  In  87  shillings  how  many  pounds  1 

23.  In  11.5  shillings  how  many  pounds  ? 

24.  In  178  shillings  how  many  pounds  ? 

25.  In  253  shillings  how  many  pounds  ? 

26.  In  6,247  shillings  how  many  pounds  1 

27.  In  38  pence  how  many  shillings  ? 

28.  In  153  pence  how  many  shillings  ? 

29.  In  1,486  pence  how  many  shillings'? 

30.  In  26,842  pence  how  many  shillings  1 

31.  In  89  farthings  how  many  pence  ? 

32.  In  243  farthings  how  many  }>ence  ? 

33.  In  3,764  farthings  how  many  pence  1 

34.  In   137   farthings   how   many   pence  1     How   manj 
shillings  1 

35.  In  382  farthings  how  many  shillings  1 


34  ARITHMETIC.  Part  1. 

36.  In  370  pence  how  many  shillings  1  How  many 
pounds  1 

37.  In  846  pence  how  many  pounds  ? 

38.  In  3,858  pence  how  many  pounds  1 

39.  In  2,340  farthings  how  many  pence?  How  many 
shillings  ?     How  many  pounds  ? 

40.  In  87,253  farthings  how  many  pounds  ? 

41.  In  87  pints  how  many  quarts  1     How  many  gallons  1 

42.  In  230  pints  how  many  gallons  ? 

43.  In  9S  gills  how  many  pints  1     How  many  quarts  1 

44.  In  183  gills  how  many  pints  1  How  many  quarts  1 
How  many  gallons  ? 

45.  In  4,217  gills  how  many  quarts  1    How  many  gallons  1 

46.  In  23,864  gills  how  many  gallons  1 

47.  In  148  gallons  how  many  hogsheads  ? 

48.  In  3,873  gallons  how  many  p'pes  ?    How  many  tuns  ? 

49.  In  48,784  gills  of  wine  how  many  hogsheads  ?  How 
many  pipes  1    How  many  tuns  ? 

50.  In  873  seconds  how  many  minutes  1 

51.  In  87  hours  how  many  days  ? 

52.  In  73  days  how  many  weeks  ?      How  many  months  ? 

53.  In  2,738  minutes  how  many  hours  ?  How  many  days  ? 
54    In  24,796,800  seconds  how    many  minutes?     How 

many  hours  ?    How  many  days  ?    How  many  weeks  ?    How 
many  months  ? 

^o.  In  506,649,600  seconds  how  many  years,  allowing 
365  days  to  the  year  ? 

56.  In  273  drams  how  many  pounds  Avoirdupois  ? 

57.  In  5,079  drams  how  many  ounces  ?  How  many 
pounds  ? 

58.  In  573,440  drams  how  many  ounces  ?  How  many 
pounds  ?  How  many  quarters  ?  How  many  hundred- 
weight ?     How  many  tons  ? 

59.  In  5,592,870  ounces  how  many  tons  ? 

60.  In  384  grains  Troy  how  many  prnny-weights  ? 

61.  In  325  dwt.  how  many  ounces  ? 

62.  In  431  oz.  Troy  how  many  pounds  ? 

63.  In  198,706  grains  Troy  how  many  penny-weights  ? 
How  many  ounces  ?    How  many  pounds  ? 

64    In  678,418  grains  Troy  how  many  pounds  1 

65.  In  37  nails  how  many  yards  ? 

GQ.  In  87  nails  how  many  ells  English  1 


IX.  DIVISION.  35 

67.  In  243  nails  how  many  yards  1 

68.  In  372  quarters  how  many  ells  Flemish  1 
&}.   In  3,107  nails  how  many  ells  Flemish  ? 

70.  In  327  shillings  how  many  English  guineas? 

71.  In  68  pence  how  many  six-pences  ? 

72.  In  130  pence  how  many  eight-pences  ? 

73.  In  342  pence  how  many  four-pences  ? 

74.  In  2,086  pence  how  many  nine-pences  ? 

75.  In  3,876  half-pence  how  many  pence  ? 

76.  In  3,948  farthings  how  many  pence  ?  How  many 
three-pences  ? 

77.  In  58,099  half-pence  how  many  pounds  ? 

78.  In  57,604  farthings  how  many  guineas  at  28s.  each  ? 

79.  In  3c£.  how  many  pence  ?    How  many  three-pences  1 

80.  In  73c£.  how  many  shillings  1  In  these  shillings  how 
many  dollars  ? 

81.  In  84-£.  how  many  shillings  1  In  these  shillings  how 
many  guineas  ? 

82.  In  37o£'.  4s.  how  many  shillings  1  How  many  dollars  1 

83.  How  many  pence  are  there  in  a  dollar  ? 

84.  In  382  pence  how  many  dollars  ? 

85.  In  \^2£.  8'.  4d.  how  many  dollars  1 

86.  In  13  yards  how  many  quarters?  In  these  quarters 
how  many  ells  Flemish  ? 

87.  In  2  y.  3  qr.  how  many  quarters  1  In  these  quarters 
how  many  ells  English? 

88.  In  17  ells  Flemish  how  many  quarters?  In  these 
quarters  how  many  aunes  ? 

89.  In  73  aunes  how  many  yards  ? 

90.  From  Boston  to  Liverpool  is  about  3,000  miles;  if  a 
sliip  sail  at  the  rate  of  1 15  miles  in  a  day,  in  how  many  days 
will  she  sail  from  Boston  to  Liverpool  ? 

91.  If  an  ingot  of  silver  weigh  36  oz.  10  dwt.  how  many 
pence  is  it  worth  at  3d.  per  dwt.  ?     How  many  pounds  ? 

92.  How  many  spoons,  weighing  17  dwt.  each,  may  oe 
made  of  31b.  6  oz.  18  dwt.  of  silver  ? 

93.  A  goldsmith  sold  a  tankard  foi  10£.  8s.  at  the  rate  of 
5s.  4d.  per  ounce.     How  muoh  did  ii  weigh  ? 

94.  How  many  coats  may  be  made  of  47  yds.  1  qr.  of 
broadcloth,  allowing  1  yd.  3  qrs.  to  a  coat  ? 

95.  What  number  of  bottles,  containing  1  pt.  2  gls.  each, 
may  be  filled  with  a  barrel  of  cider? 

96.  How  many  vessels,  containing  pints,  quarts,  and  two 


36  ARITHMETIC.  Part  1. 

quarts,  and  of  each  an  equal  number,  may  be  filled  with  a 
pipe  of  wine  1 

Note.  Three  vessels,  the  first  containing  a  pint,  the  se- 
cond a  quart,  and  the  third  two  quarts,  are  the  same  as  one 
vessel  containing  3  qts.  1  pt.  The  question  is  the  same  as 
if  it  had  been  asked,  how  many  vessels,  each  containing  3 
qts.  1  pt.,  might  be  filled. 

97.  A  man  hired  some  labourers,  men  and  boys,  and  of 
each  an  equal  number  ;  to  the  men  he  gave  7s.  and  to  the 
boys  3s.  a  day,  each.  How  many  shillings  did  it  take  to 
pay  a  man  and  a  boy  ?  It  took  3^.  10s.  to  pay  them  for  1 
day's  work.     How  many  were  there  of  each  sort  ? 

Note,  The  question  is  the  same  as  if  it  were  asked,  how 
many  men  this  money  would  pay  at  10s.  per  day. 

98.  A  man  bought  some  sheep  and  some  calves,  and  of 
each  an  equal  number,  for  6165.00;  for  the  sheep  he  gave 
67.75  apiece,  and  for  the  calves  $3.25.  How  many  were 
there  of  each  sort  ? 

99.  A  man  having  $70.15,  wished  1o  purchase  some  rye, 
some  wheat,  and  some  corn,  and  an  equal  number  of  bushels 
of  each  kind.  The  rye  was  $0.95  per  bushel,  the  wheat 
$1.37,  and  the  corn  $0.73.  How  many  bushels  of  each  sort 
could  he  buy  if  he  laid  out  all  his  money  1 

100.  How  many  table  spoons,  weighing  23  dwt.  each,  and 
tea  spoons,  weighing  4  dwt.  G  gr.  each,  and  of  each  an  equal 
number,  may  be  made  from  41b.  1  oz.  1  dwt.  of  silver  ? 

101.  A  merchant  has  20  hhds.  of  tobacco,  each  contain- 
ing 8  cwt.  3  qrs.  14  lb.  which  he  wishes  to  put  into  boxes 
containing  71b.  each.     How  many  boxes  must  he  get  ? 

102.  Bought  140  hhds  of  salt,  at  $4.70  per  hlid.  ;  how 
much  did  it  come  to  ?  How  many  quintals  of  fish,  at  $2.00 
per  quintal,  will  it  take  to  pay  for  it  ? 

103.  A  man  bought  18  cords  of  wood,  at  8  dollars  a  cord, 
and  paid  for  it  with  flour,  at  $6  a  barrel.  How  many  bar- 
rels did  it  take  ? 

104.  A  man  sold  a  hogshead  of  molasses  at  $0.40  per 
gal.,  and  received  his  pav  in  corn  at  $0.84  per  bushel.  How 
many  bushels  did  he  receive  ? 

105.  How  much  coflTee.  at  $0.25  a  pound,  can  I  have  for 
iOO  lb.  of  tea,  at  $0.87  per  lb.  -» 


IX.  ARITHMETIC.  37 

106.  How  much  broadcloth,  at  $6.6(5  per  yard,  must  be 
given  for  2  hhcls.  of  molasses,  at  $0.J}7  per  gal.  ? 

107.  How  many  times  is  8  contained  in  6,8481 

108.  12,873  is  how  many  times  3  ? 

109.  86,436  is  how  many  times  9  ? 
]  10.   1,740  is  how  many  times  6  ? 

111.  18,345  is  how  many  times  5  1 

1 12.  64,848  is  how  many  times  4  1 

113.  94,456  is  how  many  times  8  1 

114.  80,055  is  how  many  times  15? 

115.  8,772  is  how  many  times  12  1 

116.  1,924  is  how  many  limes  37? 

1 17.  1,924  is  how  many  times  52  ? 

118.  3,102  is  how  many  times  94  1 

119.  3,102  is  how  many  times  33  1 

120.  4,978  is  how  many  times  131  ? 

121.  28,125  is  how  many  times  375  I 

122.  15,341  is  how  many  times  529  1 

123.  49,640  is  how  many  times  136? 

124.  6,81  j,978  is  how  many  times  8,253  ? 

125.  92,883,780  is  how  njanv  times  9,876  ? 
12().  2,001,049,068  is  how  niany  times  261,986? 

127.  1 1,714,545,304  is  how  many  times  87,362  ? 

128.  921.253,442,978,025  is  how  many  times  918,273,645 1 

Miscellaneous  Examples. 

1.  At  4s.  3d.  per  bushel,  what  cost  3  bushels  cf  corn  ? 

2.  At  2s.  3d.  per  yard,  what  cost  4  yaids  of  cloth  ? 

3.  What  cost  7  lb.  of  coffee,  at  Is.  6d.  per  lb  ? 

4.  What  cost  3  gallons  of  wine,  at  8s.  3d.  per  gal.  ? 

5.  What  cost  4  quintals  offish,  at  13s.  3d.  per  quintal  1 

6.  What  cost  5  cwt.  of  iron,  at  \£.  9s.  4d.  per  cut.  ? 

7.  What  cost  6  cwt.  of  sugar,  at  3^.  8s.  4d.  per  cwt.  ? 

8.  What  cost  9  yds.  of  broadcloth,   at  2£.  6s.  8d.  per 
yard  ? 

9.  How  much  sugar  in  3  boxes,  each  box  containino:  14 
lb.  /  oz.  ? 

HK  At  3^.  9s.  per  cwt.  what  cost  7  cwt.  of  wool  ? 

1 1.  What  is  the  value  of  5  cwt.  of  raisins,  at  2=^.  Is.  8d. 
per  cwt.  ? 

12.  How  much  vvool  in  3  packs,  each  pack  weighing  2 
cwt.  2  qrs.  13  lb.  ? 

4 


3S  ARITHMETIC.  Part  1. 

13.  What  is  the  weight  of  5  casks  of  raisins,  each  cask 
M'eighing  2  cwt.  3  qrs.  25  lb.  ? 

14.  What  is  the  weight  of  12  pockets  of  hops,  each  pock- 
et weighing  1  cwt.  2  qrs.  17  lb.  ] 

15.  What  is  the  weight  of  16  pigs  of  lead,  each  pig  weigh- 
ing 3  cwt.  2  qrs.  17  lb.  1 

Note.  Divide  the  multiplier  into  factors  as  in  Art.  IV.  ; 
that  is,  find  the  weight  of  4  pigs  and  then  of  16. 

16.  At  7s.  4d.  per  bushel,  what  cost  18  bushels  of  wheat? 

17.  What  cost  21  cwt.  of  Iron,  at  1^.  6s.  8d.  per  cwt.  1 

18.  What  cost  28  lb.  of  tea,  at  5s.  7d.  per  lb.  ? 

19.  What  cost  32  lb.  of  coffee,  at  Is.  8d.  per  lb.  ? 

20.  What  cost  23  lb.  of  tea,  at  4s.  3d.  per  lb.  ? 

Note.  Find  the  price  of  21  lb.  and  then  of  2  lb.  and  add 
them  together.  Art.  IV. 

21.  What  cost  26  yds.  of  cloth,  at  8s.  9d.  per  yd.  ? 

22.  What  cost  34  cwt.  of  rice,  at  l£.  Is.  8d.  per  cwt.  ? 

23.  If  an  ounce  of  silver  cost  6s.  9d.,  what  is  that  per  lb. 
Troy  I     What  would  2  lb.  7  oz.  cost  1 

24.  What  is  the  value  of  38  yds.  of  cloth,  at  ^£.  6s.  4d. 
per  yd.  ? 

25.  A  man  bought  a  bushel  of  corn  for  5s.  3d.,  and  a 
bushel  of  wheat  for  7s.  6d.  ;  what  did  the  whole  amount  to  ? 

26.  How  much  silver  in  6  table  spoons,  each  weighing  5 
oz.  10  dwts.  1 

27.  A  man  bought  two  loads  of  hay,  one  weighing  18 
cwt.  3  qrs.,  and  the  other  19  cwt.  1  qr.  ;  how  much  in  both  1 

28.  A  man  bought  one  load  of  hay  for  7^.  3s.,  and 
another  for  6^.  8s.  4d. ;  how  much  did  he  give  for  both  ? 

29.  A  man  bought  3  vessels  of  wine  ;  the  first  contained 
18  gallons  ;  the  second  15  gals.  3  qts. ;  and  the  third  17 
gals.  2  qts.  1  pt.     How  much  in  the  3  vessels  1 

30.  A  merchant  bought  4  pieces  of  cloth.  The  first  con- 
tained 18  yds.  3  qrs. ;  the  second  23  yds.  1  qr.  3  nls. ;  the 
third  25  yds. ;  and  the  fourth  16  yds.  2  qrs.  2  nls.  How 
many  yards  in  the  whole  ? 

31.  A  man  bought  3  bu.  2  pks.  of  wheat  at  one  time  ;  18 
bu.  3  pks.  at  another  time  ;  9  bu.  1  pk.  5  qts.  at  a  third  ; 
and  16  bu.  0  pk.  7  qts.  at  a  fourth.  How  many  bushels  did 
he  buy  in  the  whole  ? 

32.  A  man  bought  a  cask  of  raisins  for  \£.  18s.  4d. ;  1 
lb.  of  coffee  for  Is.  6d. ;  1  cwt.  of  cocoa  for  3£.  17s. ;  1  keg 


IX.  ARITHMETIC.  39 

of  molasses  for  13s.  7d.  ;  1  box  of  lemons  for  l£.  3s. ;  1 
bushel  of  corn  for  4s.  3d.  How  much  did  the  whole  amount 
to? 

33.  A  man  bought  4  bales  of  cotton.  The  first  contained 
4  cwt.  2  qrs.  16  lb. ;  the  second  3  cwt.  1  qr.  14  lb. ;  the 
tliird  5  cwt.  0  qr.  23  lb. ;  and  the  fourth  4  cwt.  3  qrs.  What 
was  the  weight  of  the  whole  ? 

34.  A  merchant  bought  a  piece  of  cloth,  containing  19 
yds.  3  qrs.,  and  sold  4  yds.  1  qr.  of  it ;  how  much  had  he 
left  ? 

35.  A  grocer  drew  out  of  a  hhd.  of  wine  17  gals.  3  qts. ; 
how  much  remained  in  the  hogshead  1 

36.  A  bought  of  B  a  bushel  of  wheat  for  7s.  6d.  He  gave 
him  1  bushel  of  corn  worth  5s.  3d.  and  paid  the  rest  in 
money.     How  much  money  did  he  pay  1 

37.  C  bought  of  B  a  bale  of  cotton  for  18^.  4s.  and  B 
bought  of  C  4  barrels  of  flour  for  9^.  3s.  C  paid  B  the  rest 
in  money.     How  much  money  did  he  pay  ? 

38.  If  from  a  piece  of  cloth,  containing  9  yds.  you  cut  off 
I  yd.  1  qr.,  how  much  will  there  be  left  ? 

39.  If  from  a  piece  of  cloth,  containing  18  yds.  1  qr.  you 
cut  off  3  yds.  3  qrs.,  how  much  will  be  left  1 

40.  If  from  a  box  of  butter,  containing  15  lb.  there  be 
taken  61b.  3  oz.,  how  much  will  be  left  ? 

41.  A  man  sold  a  box  of  butter  for  17s.  4d.,  and  in  pay 
received  7  lb.  of  sugar,  worth  9d.  2qr.  per  lb.  and  the  rest  in 
money.     How  much  money  did  he  receive  ? 

42.  A  countryman  sold  a  load  of  wood  for  2^.  8s.  and 
received  in  pay  3  gals,  of  molasses  at  2s.  3d.  per  gal.,  8  lb. 
of  raisins  at  lOd.  per  lb.,  1  gal.  of  wine  at  lis.  3d.,  and  the 
rest  in  money.     How  much  money  did  he  receive  ? 

43.  A  smith  bought  17  cwt.  3  qrs.  of  iron,  and  after  hav- 
ing wrought  a  few  days,  wishing  to  know  how  much  of  it  he 
had  wrought,  he  weighed  what  he  had  left,  and  found  he  had 
8  cwt.  1  qr.  13  lb.     How  much  had  he  wrought  ? 

44.  A  merchant  bought  110  bars  of  iron,  weighing  53 
«wt.  1  qr.  11  lb.,  of  which  he  sold  19  bars,  weighing  9  cwt. 
3  qrs.  15  lb.     How  much  had  he  left  1 

45.  A  merchant  bought  17  cwt.  2  qrs.  1  lb.  of  sugar,  and 
sold  13  cwt.  3  qrs.  17  lb.     How  much  remains  unsold  1 

46.  From  a  piece  of  cloth,  which  contained  43  yds.  1  qr., 
a  tailor  cut  3  suits,  containing  6  yds.  2  qrs.  2  nls.  sach. 
How  much  cloth  was  there  left '? 


40  ARITHMETIC.  Part  L 

47.  The  revolutionary  war  between  England  and  Ameri- 
ca commenced  April  19th,  1775,  and  a  general  peace  took 
place  Jan.  20th,  1783.     How  long  did  the  war  continue  1 

48.  The  war  between  England  and  the  United  States 
commenced  June  18th  1812,  and  continued  2  years  8 
months   18  days.     When  was  peace  concluded  ? 

49.  The  transit  of  Venus  (that  is,  Venus  appeared  to  pass 
over  the  sun)  A.  D.  1769,  took  place  at  Greenwich,  Eng. 
June  4th,  5  h.  20  min.  50  sec.  morn.  Owing  to  the  differ- 
ence of  longitude  between  London  and  Boston  it  would  take 
place  4  hours  44  min.  16  sec.  earlier  by  Boston  time.  At 
what  time  did  it  take  place  at  Boston  1 

X.  1.*  If  I  yard  of  c!oth  is  worth  2  dollars,  what  is  ^  of 
a  yard  worth  ? 

2.  What  is  i  of 2  dollars? 

3.  If  2  dollars  will  buy  1  lb.  of  indigo,  how  much  will  1 
dollar  buy  ?  How  much  will  3  dollars  buy  ?  How  much 
will  7  dollars  buy  ?  How  much  will  23  dollars  buy  1  How 
much  will  125  dollars  buy. 

4.  At  3  shillings  per  bushel,  what  will  ^  of  a  bushel  of 
corn  cost  ?     What  will  |  of  a  bushel  cost  ? 

5.  At  3  dollars  a  barrel,  what  part  of  a  barrel  of  cider  will 
I  dollar  buy  ?  What  part  of  a  barrel  will  2  dollars  buy  1 
How  much  will  4  dollars  buy  ?  How  much  will  5  dollars 
buy  ?  How  much  will  8  dollars  buy  1  How  much  will  28 
dollars  b  ly  ? 

6.  At  3  dollars  a  box,  how  many  boxes  of  raisins  may  be 
bought  for  125  dollars  ? 

7.  How  many  bottles,  holding  3  pints  each,  may  be  filled 
with  85  gallons  of  cider  ? 

8.  At  4  dollars  a  yard,  how  much  will  i  of  a  yard  of  cloth 
cost  1  How  much  will  |  of  a  yard  cost  ?  How  much  will 
f  of  a  yard  cost  ? 

9.  A  4  dollars  a  box,  what  part  of  a  box  of  oranges  may 
be  bought  for  1  dollar  ?  What  part  for  2  dollars  ?  What 
part  for  3  dollfirs  ?  How  many  boxes  may  be  bought  for  5 
dollars  ?     How  many  for  19  uollars  1 

10.  At  4  dollars  a  barrel,  how  many  barrels  of  rye  fiour 
may  be  bought  for  327  dollars  ? 

11.  At  5  dollars  a  cord,  what  will  y  of  a  cord  of  woo«S 

*  See  Fiist  Lessons,  sect.  III.  art.  B 


X.  ARITHMETIC.  41 

cost  ?     What  will  |  cost  1     What  will  |  cost  1     What  will 
}  cost  ?    What  will  |  cost  ?     What  will  f  cost  1 

12.  At  5  dollars  a  week,  what  part  of  a  week's  board  can 
I  have  for  1  dollar  1  What  part  for  2  dollars  1  What  part 
for  3  dollars  ?  What  part  for  4  dollars  1  How  long  can  I  be 
boarded  for  7  dollars  ?  How  long  for  18  dollars  1  How  long 
for  39  dollars  ? 

13.  At  5  dollars  a  barrel,  how  many  barrels  of  fish  may  be 
bought  for  $453  ? 

14.  If  a  firkin  of  butter  cost  6  dollars,  how  much  will  J- 
of  a  firkin  cost  ?  How  much  will  |  cost  ?  How  much  will 
I  cost  ?     How  much  will  ^  cost  ?     How  much  will  y  cost  1 

15.  At  6  dollars  a  ream,  what  part  of  a  ream  of  paper  ir».ay 
be  bought  for  1  dollar  ?  What  part  for  2  dollars  ?  What 
part  for  5  dollars  ?  How  many  reams  may  be  bought  for  17 
dollars  ?     How  many  will  56  dollars  buy  ? 

16.  At  6  dollars  a  barrel,  how  many  barrels  of  flour  may 
be  bought  for  437  dollars  ? 

17.  If  a  stage  runs  at  the  rate  of  7  miles  in  an  hour,  in 
what  part  of  an  hour  will  it  run  1  mile  7  In  what  part  of  an 
hour  will  it  run  3  miles  ?  In  what  part  of  an  hour  wdl  ic  run 
5  miles  ?  In  what  time  will  it  run  17  miles  1  In  what  time 
will  it  run  59  miles  ?  In  what  time  will  it  run  from  Boston 
to  New  York,  it  being  250  miles  ? 

18.  At  8  dollars  a  chaldron,  how  many  chaldrons  of  coals 
may  be  bought  for  75  dollars  1 

19.  At  5  dollars  a  ream,  how  many  reams  of  paper  may 
be  bought  for  253  dollars  1 

20.  At  7  dollars  a  barrel,  how  many  barrels  of  flour  may 
be  bought  for  2,434  dollars  ? 

21.  At  9  dollars  a  barrel,  how  many  barrels  of  beef  may 
be  bought  for  3,827  dollars  1 

22.  At  8  dollars  a  cord,  how  many  cords  of  wood  may  be 
bought  for  853  dollars  ? 

23.  At  17  cents  per  lb.,  how  many  pounds  of  chocolate 
may  be  bought  for  $1.00  ?  How  many  lb.  for  $2.00  1  How 
many  lb.  for  $8.87  ? 

24.  At  25  dollars  per  cwt.  what  part  of  1  cwt.  of  cocoa 
may  be  bought  for  1  dollar  7  What  part  for  3  dollars  ?  What 
part  for  8  dollars  ?  What  part  for  18  dollars  ?  How  many 
cwt.  may  be  bought  for  2,387  dollars  1 

25.  At  28  dollars  per  ton,  how  many  tons  of  hay  may  be 
bought  for  $427  1 

4* 


42  ARITHMETIC.  Part  1. 

20.  If  32  dollars  will  buy  1  thousand  of  staves,  what  part 
of  a  thousand  may  be  bought  for  I  dollar  ?  What  part  of  a 
thousand  may  be  bought  for  2  dollars  1  What  part  of  a 
thousand  may  be  bought  for  7  dollars  ?  What  part  for  15 
dollars  ?  What  part  for  27  dollars  ?  How  many  thousands 
may  be  bought  for  87  dollars  ?      How  many  for  8853  ? 

27.  At  45  cents  per  gallon,  what  part  of  a  gallon  may  be 
bought  for  1  cent  ?  What  part  for  3  cents  ?  What  part  for 
8  cents  ?  What  part  for  17  cents  1  What  part  for  37  cents  1 
W^hat  part  for  42  cents  ?  How  many  gallons  may  be  bought 
for  §17.53? 

2:!<.  At  138  dollars  per  ton,  what  part  of  a  ton  of  potash 
may  be  bought  for  1  dollar?  Vv^hat  part  for  17  dollars? 
What  part  for  35  dollars  ?  What  part  for  87  dollars  ?  What 
part  for  1 15  dollars  ?  How  many  tons  may  be  bought  for 
$875  ?    How  many  tons  for  §27,484  ? 

29.  At  86.75  per  barrel,  what  part  of  a  oarrel  of  flour 
may  be  bought  for  1  cent?  What  part  for  17  cents  ?  What 
part  for  87  cents  ?  What  part  for  §2.87  ?  How  many  bar- 
![els  may  be  bought  for  873.25  ? 

30.  At  73  cents  a  gallon,  how  many  gallons  of  wine  may 
be  bought  for  §35.00  ? 

31.  At  §2.75  per  cwt.,  how  many  cwt.  of  fish  may  be 
bouglit  for  §93.07  ? 

32.  If  a  ship  sail  at  the  rate  of  132  miles  in  a  day,  in 
how  many  days  will  she  sail  3,000  miles  ? 

33.  If  a  ship  sail  at  the  rate  of  125  miles  per  day,  how 
long  will  it  take  her  to  sail  round  the  world,  it  being  about 
24,911  miles? 

34.  How  much  indigo,  at  2  dollars  per  lb.,  must  be  given 
for  19  yds.  of  broadcloth,  at  7  dollars  per  yard  ? 

35.  How  many  bushels  of  corn,  at  5s.  per  bushel,  must  be 
given  for  23  bushels  of  wheat,  at  7s.  per  bushel  ? 

30.  How  many  lb.  of  butter,  at  23  cents  per  lb.  must  be 
given  for  5  quintals  offish,  worth  §2.25  per  quintal  ? 

37.  How  many  bushels  of  potatoes,  at  3s.  per  bushel,  must 
be  given  for  a  barrel  of  flour,  worth  7  dollars  and  4  shil- 
lings ? 

38.  At  2£.  3s.  per  barrel,  how  many  shillings  will  7  bar- 
rels of  flour  come  to  ?  How  much  brandy,  at  8s.  per  gal., 
will  it  take  to  pay  for  it  ? 

39.  If  03  gallons  of  water,  in  1  hour,  run  into  a  cistern 
containing  423  gallons,  in  what  time  will  it  be  filled  1 


XI.  ARITHMETIC.  43 

40.  At  4s.  3d.  per  bushel,  what  part  of  a  bushel  will  Id. 
buy  ?  What  part  of  a  bushel  will  8(J.  buy  1  What  part  of  a 
bushel  will  Is.  or  I2d.  buy  ?  How  many  bushels  may  be 
bought  for  2£.  l(3s.  4d  ? 

41.  At  8s.  4d.  per  gallon,  how  many  gallons  of  wine  may 
be  bought  for  \7£.  3s.  8d.  ? 

42.  At  lis.  Od.  per  gallon,  how  many  gallons  of  brandy 
may  be  bought  for  43ii'.  ? 

43.  A  buys  of  B  3  cwt.  3  qrs.  of  sugar,  at  9  cents  per 
lb. ;  2  hhds.  of  brandy  at  81-57  per  gallon  ;  and  8  qqls.  of 
fish  at  $2.55  per  qql.  In  return,  B  pays  A  $25.00  in  cash ; 
150  lb.  of  bees-wax,  at  80.40  per  lb. ;  and  the  rest  in  flour 
at  $7.50  per  barrel.  How  many  barrels  of  flour  must  B 
give  A  ? 

44.  785  are  how  many  times  4 1 

45.  2,873  are  how  many  times  8 1 

46.  8,467  are  how  many  times  9  ? 

47.  2,864  are  how  many  times  14  1 

48.  43,657  are  how  many  times  28  ? 

49.  27,647  are  how  many  times  78  ? 
60.  884,673  are  how  many  times  153  t 

51.  181,700  are  how  many  times  437  1 

52.  984,607  are  how  many  times  2,4671 

53.  Divide  1,708,540  by  13,841. 

54.  Divide  407,648,205  by  403,006. 

55.  Divide  100,000,000  by  12,478. 

XI.     1.  At  10  cents  per  lb.,  how  many  lb.  of  beef  may  be 

bought  for  $0.87  ? 

2.  At  10  cents  per  lb.  how  many  lb.  of  cheese  may  be 
bought  for  $3.54  ? 

3.  At  lOd.  per  lb.  how  many  lb.  of  raisins  may  be  bought 
for  13s.  4d.  ? 

4.  Suppose  you  had  243  lb.  of  candles,  which  you  wished 
to  put  into  boxes  containing  10  lb.  each  ;  how  many  boxes 
would  they  fill  ? 

5.  At  10  dollars  a  chaldron,  how  many  chaldrons  of  coal 
may  be  bought  for  749  dollars  1 

6.  At  $1.00  per  bushel,  how  many  bushels  of  corn  can 
you  buy  for  $43.73  ? 

7.  If  you  had  32,487  oranges,  which  you  wished  to  put 
into  boxes  containing  100  each,  how  many  boxes  could  you 
fiU? 


44  ARITHMETIC.  Part  1, 

8.  At  $1.00  per  lb.  how  many  lb.  of  hyson  tea  may  be 
bought  for  $243.84  1 

9.  At  $10.00  per  bbl.  how  many  barrels  of  pork  may  be 
bought  for  $247.63  ? 

10.  At  $100.00  per  ton,  how  many  tons  of  iron  may  be 
bought  for  $8,734.87  ? 

11.  In  78  how  many  times  10  ? 

12.  In  3,876  how  many  times  10  ? 

13.  In  473  how  many  times  100? 

14.  In  6,783  how  many  times  100? 

15.  In  48,768  how  many  times  100  ? 

16.  In  47-5,384  cents  how  many  dollars  ? 

17.  In  5,710,648  how  many  times  1,000  ? 

18.  In  1,764,874  mills  how  many  cents?     How  many 
dimes  ?     How  many  dollars  ? 

19.  In  4,710,074  mills  how  many  dollars  ? 

XII.     I.  What  part  of  5  lb.  is  3  lb.  ? 

2.  What  part  of  7  yards  is  4  yards  ? 

3.  What  part  of  7  yards  is  10  yards? 

4.  What  part  of  3  yards  is  5  yards  ? 

5.  What  part  of  4  oz.  is  7  oz.  ? 

6.  What  part  of  7d.  is  lOd.  ? 

7.  What  part  of  17  cents  is  9  cents  ? 

8.  What  part  of  9  cents  is  17  cents  ? 

9.  What  part  of  35  dollars  is  17  dollars  ? 

10.  What  part  of  17  dollar's  is  35  dollars  ? 

11.  4  dollars  is  what  part  of  67  dollars? 

12.  67  dollars  is  what  part  of  4  dollars? 

13.  What  part  of  103  rods  is  17  rods  ? 

14.  What  part  of  17  rods  is  103  rods  ? 

15.  What  part  of  256  miles  is  39  miles  ? 

16.  What  part  of  39  miles  is  256  miles  ? 

17.  What  part  of  287  inches  is  138  inches  ? 

18.  What  part  of  38,649  farthings  is  8,473  farthings? 

19.  What  part  of  907,384  is  3,906  ? 

20.  What  part  of  384  is  96,483  ? 

21.  What  part  of  Id.  is  1  farthing  ?     What  part  of  Id.  is 
2  farthings  ?  3  farthings  ? 

22.  What  part  of  Is.  is  Id.  1  2d.  ?  3d.  ?  4d.  ?  5d.  ?  6d.  ? 
7d.?  lid.  ?  .  ■ 

23.  What  part  of  Is.  is  1  farthing  ?    2  farthings  ?    3  far- 
thkigs  ?  7  farthings  7  13  farthings  ?  35  farthings  ? 


Xll.  ARITHMETIC.  45 

24.  What  part  of  Is.  is  Id.  ^  qr.  ?  2d.  Iqr.  ?  9d.  2qr.  1 
Note.     Reduce  the  pence  to  farthings. 

25.  What  part  of  li*.  is  1  shilling  1  2  shillings  1  7  shiU 
lings  1   17  shillings  ? 

2().  What  part  of  \£.  is  I  penny  ?  3  pence  ?  7  pence  1 
25  pence  I  87  pence  ?   147  'lence  1 

27.  What  part  of  \£.  is  2s.  r,d.  ? 
Note.     Reduce  the  shillings  to  pence. 

28.  What  part  of  I  i:.  is  7s.  4d  ? 

29.  What  part  of  \£.  is  l:?s.  8d.  ? 

30.  What  part  of  !£.  is  I8s.  lid.? 

31.  How  many  farthings  are  there  in  }£.1 

32.  What  part  of  l£.  is  1  farthing?  3  farthings?  7  far- 
things ?  18  farthings  ?  53  farthings  ?  137  farthings  ?  487 
farthings  ? 

33.  What  part  of  l£.  is  7d.  3qr.  ? 

34.  What  part  of  l£.  is  1  Id.  2  qr.  ? 

35.  What  part  of  l£.  is  4s.  7d.  1  qr.  ? 

Note.     Reduce  the  shillings  and  pence  to  farthings. 

3().  What  part  of  \£.  is  I3s.  8d.  2qr.  ? 

37.  What  part  of  a  gallon  is  1  quart  ? 

38.  What  part  of  a  gallon  is  1  pint  ? 

39.  What  part  of  a  gallon  is  I  gill  ? 

40.  What  part  of  a  gallon  is  7  gills  ? 

41.  What  part  of  a  gallon  is  2  qts.  1  pt.  3  gls.  ? 

42.  What  part  of  I  hhd.  is  1  gallon  ?    17  gallons  t 

43.  What  part  of  1  hhd.  is  1  gill  ?  43  gills  ? 

44.  What  part  of  1  hhd.  is  17  gals.  3  qts.  1  pi.  2  gills  I 

45.  What  part  of  1  qr.  is  1  lb.  ?    13  lb.  ? 

46.  What  part  of  1  lb.  is  1  oz.  Avoirdupois?    11  oz.  T 

47.  What  part  of  1  lb.  is  1  dram  ?    15  drams  ? 

48.  What  part  of  I  lb.  is  13  oz.  11  dr.  ? 

49.  What  part  of  1  qr.  is  I  dram  ?  43  drams  ? 

50.  What  part  of  1  qr.  is  17  lb.  1 1  oz.  8  dr.  ? 

51.  What  part  of  1  year  is  1  calendar  month  ?  7  months  ? 
11  months? 

52.  What  part  of  a  calendar  month  is  1  day  ?    3  days  t 
17  days  ? 

63.  What  part  of  1  hour  is  1  minute  ?   17  minutes  1 

54.  What  part  of  1  day  is  1  minute  ?    13  minutes  t 

55.  What  part  of  1  day  is  7  h.  43  min.  ? 


4G  ARITHMETIC.  Part  1. 

56    What  part  of  1  day  is  I  second  1    73  seconds  ?    258 
seconds  ? 

57.  What  part  of  1  day  is  13  h.  43  min.  57  sec.  ? 

58.  What  part  of  a  year  is  1  second,  allowing  365  days  6 
hours  to  the  year  ?  8,724  seconds  ? 

69.  What  part  of  a  year  is  123  d.  17  h.  43  min.  25  sec.  1 

60.  What  part  of  8s.  3d.  is  1  penny  1  8  pence  1  3s.  4d.  1 

61.  What  part  of  16s.  9d.  is  5s.  3d.  ? 

62.  What  part  of  a  dollar  is  43  cents  ? 

63.  What  part  of  5  dollars  is  72  cents  1 

64.  What  part  of  3c£.  is  1  shilling  ?  17  shillings  ? 

65.  What  part  of  5^.  is  one  penny  1    1 1  pence  ?  4s.  8d.  1 

66.  What  part  of  4^.  7s.  8d.  is  13s.  6d.  1 

67.  What  part  of  13£.  8s.  5d.  is  3^.  7s.  6d.  ? 

68.  What  part  of  3  yards  is  1  quarter  of  a  yard  1 

69.  What  part  of  16  yds.  1  qr.  is  7  yds.  3  qrs.  1 

70.  What  part  of  13  yds.  3  qrs.  1  nl.  is  4  yds.  3  qrs.  3 
nl6.  ? 

71.  What  part  of  2  yds.  3  qrs.  is  7  yds.  2  qrs.  ? 

72.  What  part  of  3  days  is  5  minutes  1 

73.  What  part  of  18  d.  3  h.  is  13  d.  4  h.  ? 

74.  What  part  of  5  d.  13  h.  18  min.  is  26  d.  4  h.  7  min.  X 

75.  What  part  of  43  gals.  3  qts.  1  pt.  is  27  gals.  2  qts.  ? 

76.  What  part  of  17  gals.  1  qt.  is  87  gals.  2  qts.  ? 

77.  What  pari  of  2cwt.  1  qr.  17  lb.  is  1  cwt.  3  qrs.  191b. ! 

78.  What  is  the  ratio  of  8  to  5  ? 

79.  What  is  the  ratio  of  5  to  8  ? 

80.  What  is  the  ratio  of  28  to  9  1 

81.  What  is  the  ratio  of  9  to  28  ? 

82.  What  is  the  ratio  of  117  to  96  1 
S3.  What  is  the  ratio  of  57  to  294  1 
84.  What  is  the  ratio  of  3,878  to  943  1 


XIII.  1.*  If  a  family  consume  i  of  a  barrel  of  flour  in  a 
week,  how  many  barrels  will  last  them  4  weeks?  How 
many  barrels  will  last  them  17  weeks  ? 

2.  If  i  of  a  barrel  of  cider  will  serve  a  family  1  week, 
how  many  barrels  will  serve  them  11  weeks  1  How  many 
barrels  will  serve  them  28  weeks  1 

3.  In  y  how  many  times  11     In  ^J  how  many  times  1 1 

See  First  Lessons,  Sect.  VIII.  Art.  B. 


XIV.  ARITHMETIC.  47 

4.  If  Jy  of  a  chaldron  of  coals  will  supply  a  fire  1  day^ 
how  many  chaldrons  will  supply  it  57  days  at  that  rate? 

5.  Reduce  |^  to  a  mixed  number.  I 

6.  In  \^  of  a  bushel  how  many  bushels  ?  ' 

7.  Reduce  -—  ^^  3.  mixed  number. 

8.  In  \y  of  1£.  how  many  pounds  ? 

Note.     This  question  is  the  same  as  the  following. 

9.  In  387  shillings  how  many  pounds  ? 

10.  In  ^^  of  a  shilling  how  many  shillings  ? 

11.  In  437  pence  how  many  shillings  ? 

12.  In  \^  of  a  pound  Avoirdupois,  how  many  pounds  ? 

13.  In  134  oz.  Avoirdupois  how  many  pounds  ? 

14.  In  ^-^-^  of  a  guinea  how  many  guineas  ? 

15.  In  322  shillings  how  many  guineas,  at  28  shillings 
each  1 

16.  In  \^-f^  of  a  day  how  many  days  ? 

17.  In  476  hours  how  many  days  1 

18.  In  ^|-5-7  of  an  hour  how  many  hours  ? 

19.  In  9,737  minutes  how  many  hours  ? 

20.  In  ^Ifl-^  of  a  year  how  many  years  ? 

21.  In  43,842  days  hov/  many  years,  allowing  365  days 
to  the  year  1 

22.  In  ^jj^^  of  a  year  how  many  years  1 

23.  Reduce  '^-^  to  a  mixed  number. 

24.  Reduce  ^If  ^  ^q  ^  mixed  number. 

25.  Reduce  ?^Y  to  a  mixed  number. 

26.  Reduce  ^¥A¥f  °  ^^  ^  ^i^^^l  number. 

XIV.  1.*  If  -f  of  a  cord  of  wood  will  supply  two  fires  1 
day,  how  many  days  will  a  cord  supply  them  ?  How  many 
days  will  3  cords  supply  them  1  How  many  days  will  13 
cords  supply  them  1 

2.  How  many  7ths  are  there  in  1  ?  How  many  7ths  are 
there  in  3  ?     How  many  in  13  1 

3.  If  |-  of  a  barrel  of  beer  will  serve  a  family  I  day,  how 
many  days  will  1  barrel  serve  them  ?  How  many  days  will 
11  barrels  serve  them  ?  How  many  days  wiH  13f  barrels 
serve  them  ?     How  many  days  will  43|-  barrels  serve  them  1 

4.  In  1  how  many  8ths  ?  In  7}  how  many  8ths  ?  In 
13|  how  many  8ths  ?     In  43f  how  many  8ths  ? 

5.  If  ^L^  of  a  barrel  of  flour  will  serve  a  family  1  week 

*  See  First  Lessons,  Sect.  VIII.  Art.  A. 


48  ARITHMETIC.  Part  1. 

how  many  weeks  will  2/^  barrels  serve  them  ?     How  many 
weeks  will  Kiy^  serve  them  ? 

(5.   In  13-i?j  how  many  I'>ths?  ' 

7.  If  y^y  of  a  barrel  of  Hour  will  serve  1  man  1  day,  how 
many  men  will  l-^-j  barrels  serve  ]  How  many  men  will 
43|f  barrels  serve  ? 

8.  Reduce  1-^-j  to  an  improper  fraction. 

9.  Reduce  4;J5f  to  an  improper  fraction. 

10.  In  18f  bushels  how  many  ^  of  a  bushel  ? 

11.  In  23 ,\  barrels  how  many  barrels? 

12.  In  4y''2  shillings  how  many  -^^  of  a  shilling  7  That  is, 
in  4s.  5d.  how  many  pence  1 

13.  In  S/qX  how  many  2V  of  a  pound  1  That  is,  in  8^ 
7s-  how  many  shillings  ? 

14.  In  15.}4:  days  how  many  jt  of  a  day  7 

15.  In  15  d.  1 1  h.  how  many  hours  ? 

16.  In  \lti  horn's  how  many  ,}q  of  an  hour  1 

17.  In  17  h.  43  min.  how  many  minutes  ? 

18.  In  1y\2  ^^^^-  how  many  -j\^  of  1  cwt.  ? 

19.  In  7  cwt.  37  lb.  liow  many  pounds  ? 

20.  In  182^7  ^^^'^-  how  many  ^tt  of  1  cwt.  1 


21.  In  237  ,3_  how  many  ^\ 


22.  Reduce  437^^  to  an  improper  fraction. 

23.  Reduce  032-^,^-^  to  an  improper  fraction. 

XV.  1.*  Bought  7  yards  of  cotton  cloth,  at  |  of  a  dollar 
per  yard  ;  how  many  dollars  did  it  come  to  ? 

2.  h'a  horse  consuiiie  4  of  a  bushel  of  oais  in  1  d.'.y,  how 
many  bushels  will  he  consume  in  15  days  ? 

3.  If  a  family  consume  |  of  a  barrel  of  (lour  in  a  week, 
how  many  barrels  would  they  consume  in  17  weeks? 

4.  If  i  of  a  ton  of  hay  will  keep  1  c<»vv  through  the  win- 
ter, how  many  tons  will  keep  23  cows  the  same;  timt  ? 

5.  If  a  pound  of  beeswax  cost  -^^  of  a  dollar,  how  many 
dollars  will  7  lb.  cost  ? 

(>.  If  [  lb  of  chocolate  cost  j^y  of  a  dollar,  w  liat  will  27  lb. 
cost  ? 

7.  If  one  lb.  of  candles  cost  2^  of  a  dollar,  what  will  43 
lb.  cost  I 

8.  At  ^5  of  a  dollar  a  pound     what  cost  87  lb.  of  siieatji- 
ing  copj)er  1 

*  Sen  First  Lessons,  Seel.  IX.  An.  A 


XVI.  ARITHMETIC.  49 

9.  At  -||^  of  a  dollar  a  gallon,  what  will  1  hhd.  of  molasses 
cost? 

10.  At  j^o%  of  a  dollar  a  gallon,  what  cost  3  hhds  of  mo- 
lasses ? 

11.  At  yoQ-  of  a  dollar  a  gallon,  what  cost  5  hhds  of  rum  1 
12.*  At  71  dollars  per  cwt.  what  cost  5  cwt.  of  lead  ? 

13.  At  13|  dollars  per  thousand,  what  cost  8  thousand  of 
staves  ? 

14.  At  14f  dollars  per  barrel,  what  cost  23  barrels  of  fish  1 

15.  If  a  yard  of  cloth  cost  383V  shillings,  what  cost  15 
yards  ? 

IG.  If  a  barrel  of  beef  cost  54||  shillings,  what  cost  23 
barrels  ? 

17.  If  1  gallon  of  <^in  co.f.  ^^^  of  i^.  what  cost  1  hhd.  1 
IS.  At2^-f-Y£.  per  barrel,  what  cost  17  barrels  of  flour  ? 

19.  A  man  failing  in  trade  is  able  to  pay  only  |  of  a  dol- 
lar on  a  dollar,  how  much  will  he  pay  on  a  debt  of  5  dol- 
lars ?     How  much  on  53  dollars  ? 

20.  A  man  failing  in  trade  is  able  to  pay  only  f^  of  a  dol- 
lar on  a  dollar,  how  much  will  he  pay  on  a  debt  of  75  dol- 
lars  1  How  much  on  a  debt  of  15o  dolb.rs  1 

21.  Suppose  the  duties  on  tea  to  be  f^  of  a  dollar  on  1 
lb.,  what  would  be  the  duties  on  738  lb.  1 

22.  A  man  failing  in  trade  is  able  to  pay  only  ||^  of  a 
dollar  on  a  dollar,  how  much  can  he  pay  on  a  debt  of  873 
dollars  ? 

23.  How  much  is  5  times  j\  1 

24.  How  much  is  7  times  -J-r%- 1 


25.  How  much  is  17  times  2^5  '^ 

26.  How  much  is  9  times  2WV 

27.  How  much  is  35  times  -j-yi 


5  _7 

3  4, 


28.  How  much  is  237  times  ^\%  1 

29.  Multiply  ^-^  by  238. 

30.  Multiply  -j^VoT  by  1003. 

31.  Multiply  ^1^  by  5060. 

32.  Multiply  -jie^  by  607. 

XVI.  l.t  If  a  piece  of  linen  cost  24  dollars,  what  will  J 
of  a  piece  cost  1 

2.  If  3  chaldrons  of  coal  cost  36  dollars,  what  part  of  30 

*  See  First  Lessons,  Sect,  IX,  Art.  B, 
t  See  First  Lessons,  Sect.  V.  and  X. 

5 


50  ARITHMETIC.  Part  I, 

dollars  will  1  chaldron  cost  1     How  much  will  a  chaldron 
cost? 

3.  If  7  lb.  of  chocolate  cost  $1.54,  what  part  of  $1.54  will 
lib.  cost?     What  is  I  of  81.54? 

4.  If  9  yards  of  cloth  cost  126  dollars,  what  part  of  126 
dollars  will  1  yard  cost  ?     How  much  will  it  cost  per  yard  ? 

5.  If  17  chaldrons  of  coal  cost  ViOt  dollars,  what  part  of 
136  dollars  will  1  chaldron  cost  ?  What  is  -^  of  136  dollars  ? 

6.  A  ticket  drew  a  prize  of  652  dollars,  of  which  A  own- 
ed \  ;  what  was  A's  share  of  the  money  ? 

7.  A  privateer  took  a  prize  worth  36,960  dollars,  of  which 
the  captain  was  to  have  \,  the  first  mate  J^,  the  second  mate 
Jg-,  and  the  rest  was  to  be  divided  equally  among  the  crew, 
which  consisted  of  50  men ;  what  was  the  share  of  each  offi- 
cer, and  of  each  sailor  ? 

8.  If  a  man  travel  38  miles  in  a  day,  how  far  will  he 
travel  in  7|  days  ? 

9.  At  $2.48  per  barrel,  what  will  5i  barrels  of  cider  cost  ? 

10.  At  $1.3S  a  bushel,  wliat  will  8|  bushels  of  rye  cost  ? 

11.  At  $1.83  per  bushel,  what  will  |  of  a  bushel  of  wheat 
cost  ?     What  will  |  cost  ? 

12.  At  $7.23  per  barrel,  what  cost  4|  barrels  of  flour  ? 

13.  At  $1.92  per  gallon,  what  cost  ^  of  a  gallon  of  bran- 
dy ?    That  is,  what  cost  1  quart  ? 

14.  At  $4.20  per  box,  what  cost  ^  of  a  box  of  oranges  ? 
What  cost  J  of  a  box  ?     What  cost  1|  box  ? 

15.  At  $2.20  per  lb.,  what  cost  |  of  a  lb.  of  indigo  ? 
What  cost  7|  lb.    ? 

16.  At  $2.25  per  quintal,  what  cost  f  of  a  qql.  of  fish  ? 
What  cost  llfqqls.? 

17.  At  $7.75  per  cwt.,  what  cost  -i  cwt.  of  sugar  ?  What 
cost  f  cwt.  ?     What  cost  |-  cwt.  ? 

18.  At  $7.25  per  cask,  what  cost  3i  casks  of  Malaga  rai- 
sins ? 

19.  At  $0.75  per  bushel.,  what  cost  18|  bushels  of  In- 
dixin  corn  ? 

20.  At  $6.78  per  barrel,  what  cost  ^  of  a  barrel  of  flour  1 
What  cost  |-  of  a  barrel  ? 

21.  At  $7.86  per  barrel,  what  cost  18f  barrels  of  flour  1 

22.  If  7  bushels  of  oats  cost  $2.94,  what  part  of  $2.94 
rill  1  bushel  cost  ?     What  is  \  of  $2.94  ? 

23.  A  man  bought  8  sheep  for  $60.24  ;  what  part  of 
$60.24  did  1  sheep  cost  ?     What  is  \  of  $60.24  ? 


XVI.  ARITHMETIC.  61 

24.  A  merchant  bought  12  barrels  of  flour  for  $82.44  ; 
what  part  of  $82.44  did  1  barrel  cost  ?  What  is  -^\  of 
$82.44  ? 

25.  A  merchant  bought  IS  hhds.  of  brandy  for  $1,092.00; 
what  part  of  $1,092.00  did  1  hhd.  cost  ?  What  did  it  cost 
per  hhd.  1 

20.  If  37  lb.  of  beef  cost  $2.96,  what  part  of  $2.90  will  1 
lb.  cost  ?     AVhat  is  ^V  of  $2.90 1 

27.  If  I  hhd.  of  rum  cost  $52.92  what  part  of  $52.92  will 
1  gallon  cost  ?     How  much  will  1  gallon  cost  ? 

28.  At  43  cents  a  gallon,  what  will  15|  hhds.  of  molasses 
come  to  ? 

29.  How  many  inches  are  there  in  a  mile  7 

MEASURE    OF    LENGTH. 


3    barley-corns  (bar.) 

make  I  mch,    marl 

ked 

m. 

12    inches 

1  foot 

ft. 

3    feet 

1  yard 

yd. 

5i  yards  or  ) 
101  feet         i 

(  1  rod 
(  or  pole 

rod. 
pol. 

40    poles 

1   furlong 

fur. 

8    furlongs 

1  mile 

ml. 

3    miles 

1  league 

I. 

60    geographical  miles, 
69i  statute  miles 

'^^  >      1  degree  nearly, 

(deg. 
ioro 

lOO'  degrees  the  circumference  of  the  earth. 

Also  4  inches 

make         1  hand 

5  feet 

1  geometrical 

pace 

0  feet 

1   fathom 

0  points 

1  line 

12  lines 

1  inch 

30>  How  many  geographical  miles  is  it  round  the  earth  ? 

31.  IIow  many  statute  miles  round  the  earth  ? 

32.  How  many  inches  in  15  miles  1 

33.  How  many  rods  round  the  earth  1 

34.  How  many  barley-corns  will  reach  round  the  earth  1 

35.  At  $25.00  per  ton,  what  will  1  cwt.  of  hay  come  to  1 
30.  If  0  horses  eat  18  bushels  of  oats  in  a  week,  what  part 

of  18  bu.  will  1  horse  eat  in  the  same  time  ?    What  part  of 
18  bu.  will  5  horses  eat  ?     What  is  f  of  18  bu.  ? 

37.  If  a  man  travel  35  miles  in  7  hours,  how  many  miles 


52  ARITHMETIC.  Part  1. 

will  he  travel  in  1  hour  1     How  many  in  12  hours  1     How 
many  in  53  hours  ? 

38.  If  a  stage  run  96  miles  in  12  hours,  how  many  miles 
will  it  run  in  15  days  5  hours,  at  that  rate,  if  it  run  12 
hours  each  day  ? 

39.  At  830.00  a  ton,  what  will  7  tons  8  cwt.  of  hay  come 
to? 

40.  A  man,  after  travelling  23  hours,  found  he  had  tra- 
velled 1 15  miles  ;  how  far  had  he  travelled  in  an  hour,  sup- 
posing he  had  travelled  the  same  distance  each  hour  ?  how 
far  would  he  travel  in  47  hours  at  that  rate  1 

41.  If  1  hhd.  20  gal.  cost  $118.69,  what  is  it  a  gallon  1 
How  much  is  it  per  hhd.  1  How  much  would  3  hhds.  17 
gal.  come  to,  at  that  rate  ? 

42.  If  18  gal.  3  qts.  of  wine  cost  $33.75,  what  is  it  a 
quart  1  What  will  1  hhd.  43  gals.  2  qts.  come  to,  at  that 
rate? 

43.  If  3  qrs.  13  lb.  of  cocoa  cost  $14..55,  what  is  it  per 
lb.  ?     How  much  will  47  lb.  come  to,  at  that  rate  ? 

44.  If  1  cwt.  3  qr.  7  lb.  of  cocoa  cost  $32.48,  what  is  it 
per  lb.  ?  What  would  be  the  price  of  3  cwt.  2  qrs.  5  lb.  at 
that  rate  ? 

45.  If  1  oz.  of  silver  be  vvorth  6s.  8d.,  what  is  that  per 
dwt.  ?  What  would  be  the  price  of  a  silver  cup,  weighing 
10  oz.  14  dwts.  ? 

46.  If  1  cwt.  3  qrs.  23  lb.  of  tobacco  cost  $54.75,  what 
will  3  cwt.  2  qrs.  5  lb.  cost  at  that  rate  1 

47.  If  6  horses  will  consun^.e  19  bu.  2  pks.  of  oats  in  3 
weeks,  how  many  pecks  will  17  horses  consume  in  the  same 
time  ?     How  many  bushels  ? 

48.  A  ship  was  sold  for  .^568,  of  which  A  owned  ■^;  what 
was  A's  part  of  the  money  ? 

49.  If  3  yds.  3  qrs.  of  broadcloth  cost  |30.00,  what  will 
7  yds.  cost  ? 

50.  If  37  yds.  of  cloth  cost  $185.00,  what  will  18|  yds. 
cost? 

51.  If  23  yds.  of  cloth  cost  $230.00,  what  will  1  qr.  cost  1 
What  will  1  ell  English  cost  ?     What  will  17|  ells  cost  ? 

52.  If  a  chest  of  Hyson  tea,  weighing  79  lb.,  cost  32^. 
lis.  9d.,  what  would  43  lb.  come  to  at  that  rate  ? 

53.  if  9  cwt.  3  qrs.  4  lb.  of  tallow  cost  $109.60,  what 
will  1  cwt.  cost  ? 

54.  If  the  distance  from  Boston  to  Providence  be  40  miles. 


XVI.  ARITHMETIC.  53 

how  many  times  will  a  carriage  wheel,  the  circumference  of 
which  is  15  ft.  0  in.,  turn  round  in  going  that  distance  ? 

55.  If  the  forward  wheels  of  a  wagon  are  14  ft.  (>  in.,  and 
the  hind  wheels  15  ft.  9  in.  in  circumference,  how  many 
more  times  will  the  forward  wheels  turn  round  than  the  hind 
wheels,  in  going  from  Boston  to  New  York,  it  being  248 
miles  ? 

56.  How  many  times  will  a  ship  97  ft.  6  in.  long,  sail  her 
length  in  the  distance  of  1,200  miles  1 

57.  If  1  bushel  of  oats  will  serve  3  horses  1  day,  how 
much  will  serve  1  horse  the  same  time  1  How  much  will 
serve  2  horses  ? 

58.  If  1  bushel  of  corn  will  serve  5  men  1  week,  kow 
much  will  serve  I  man  the  same  time  ?  How  much  will 
serve  3  men  ? 

59.  If  you  divide  1  gallon  of  beer  equally  among  5  men, 
how  much  would  you  give  them  apiece  1  If  you  divide  2 
gallons,  how  much  would  you  give  them  apiece  1  If  you  di- 
vide 3  gallons,  how  much  would  you  give  them  apiece  1  If 
you  divide  7  gallons,  how  much  would  you  give  them  apiece  1 

60.  What  is  -1  of  1  ?  What  is  |  of  2  ?  What  is  i-  of  3  7 
What  is  .}  of  7  ? 

61.  If  7  yards  of  cloth  cost  1  dollar,  what  part  of  a  dollar 
will  1  yard  cost  ]  If  7  yards  cost  2  dollars,  what  part  of  a 
dollar  would  1  yard  cost  1  If  7  yards  cost  5  dollars,  what 
part  of  a  dollar  would  1  yard  cost  ?  If  7  yards  cost  10  dol- 
lar?, what  part  of  a  dollar  will  1  yard  cost  1  How  many  dol- 
lars? 

62.  Whatis|ofl?  Whatis|of2?  of3?  of5?  of  10  1 

63.  If  you  divide  1  gallon  of  wine  equally  among  13  per- 
sons, how  much  would  you  give  them  apiece  ?  How  much 
if  you  divide  2  gallons  ?  How  much  if  you  divide  3  gallons  7 
5  gallons  1  11  gallons  ?  15  gallons  1  23  gallons  1  57  gal- 
lons ? 

64.  WhatisyVofl^  of  2?  of  3?  of  5?  of  11?  of  23? 
of  57? 

65.  If  you  divide  1  dollar  equally  among  23  persons,  what 
part  of  a  dollar  would  you  give  them  apiece  ?  If  you  divide 
2  dollars,  what  part  of  a  dollar  would  you  give  them  apiece  1 
7  dollars  ?  18  dollars  ?  34  dollars  ?  87  dollars  ?  253  dol- 
lars? 

6C^.  Whatisiofl?  of2?  of 7?  of  18?  of34?  of87? 
of  253? 

5* 


64  ARITHMETIC.  Part  1. 

67.  If  8  barrels  of  flour  cost  53  dollars,  what  is  that  a 
barrel  ?     What  will  13  barrels  cost  1 

68.  If  17  lb.  of  beef  cost  $1.43,  what  is  that  per  lb.  ? 

69.  If  57  lb.  of  raisins  cost  $8.37,  how  much  is  that  per 
lb.  1     What  will  43  lb.  cost  ? 

70.  If  1  cwt.  3  qrs.  15  lb.,  of  sugar  cost  $19..53,  how 
much  is  it  per  lb.  ?     What  will  6  cwt.  1  qr.  23  lb.  cost  1 

71.  If  15  yds.  3  qrs.  of  broadcloth  cost  $147.23,  what 
will  1  qr.  cost  ?  What  will  a  yard  cost  ?  What  will  57 
yards  cost  *? 

72.  Bought  3  hhds.  of  wine  for  8257.00  ;  what  was  it  per 
gallon  1     What  would  5  pipes  cost  at  that  rate  ? 

73.  If  2  bushels  of  wheat  is  sufficient  to  sow  3  acres,  what 
part  of  a  bushel  will  sow  1  acre  1  How  much  will  sow  5 
acres  1 

74.  If  5  barrels  of  cider  will  serve  8  men  1  year,  what 
part  of  a  barrel  will  serve  1  man  the  same  time  ?  How 
much  will  serve  17  men  ?    " 

75.  If  5  barrels  of  flour  will  serve  23  men  1  month,  what 
part  of  a  barrel  will  serve  1  man  the  same  time  1  How  much 
wfll  serve  75  men  1 

76.  If  3  acres  produce  43  bushel?  of  wheat,  what  part  of 
an  acre  will  produce  I  bushel  ?  How  much  will  produce  7 
bushels  ?  How  much  will  produce  28  bushels  ?  How  much 
will  produce  153  bushels  ? 

77.  If  7  acres  1  rood  produce  123  bushels  3  pks.  of  wheat, 
how  much  will  1  rood  produce  1  How  much  will  25  acres 
produce  1 

Note.     4  roods  make  1  acre. 

78.  If  9  acres  1  rood  produce  136  bushels  of  rye,  what 
part  of  a  rood  will  produce  1  bushel  1  How  many  acres  will 
produce  500  bushels  ? 

79.  If  435  men  consume  96  barrels  of  provisions  in  9 
months,  how  many  barrels  will  2,426  men  consume  in  the 
same  time  ? 

80.  At  23  cents  per  gallon,  what  will  |  of  a  hhd.  of  mo- 
lasses come  to  ? 

81.  At  14  cents  per  lb.,  what  will  -j  of  1  cwt.  of  raisins 
come  to  ? 

82.  How  many  shillings  in  -f  of  \£.  1 
S3.  How  many  pence  in  -^  of  a  shilling  ? 
84.  How  many  pence  in  |  of  a  shilling  % 


XVr.  ARITHMETIC.  55 

85.  How  many  farthings  in  |  of  a  penny  ? 

86.  Find  the  value  of  |  of  a  shilling,  in  pence  and  farthings. 

87.  Find  the  value  of -f  of  a  shilling,  in  pence  and  flirthings. 

88.  Find  the  value  of  |  of  l£.,  in  shillings  and  pence. 

89.  Find  the  value  of  j-  of  l£.,  in  shillings,  pence,  and 
farthings. 

90.  What  is  the  value  of  j^-:^  of  1^.,  in  shillings,  pence, 
and  farthings  ? 

91.  Find  f  of  a  day  in  hours,  minutes,  and  seconds. 

92.  Find  |  of  I  hour  in  minutes  and  seconds. 

93.  What  is  y%  of  a  day  1 

94.  What  is  2^  of  a  day  1 

95.  What  is  f  of  1  lb.  Avoirdupois  ? 

96.  What  is  ^  of  1  cwt.  in  quarters  and  lb.  1 

97.  What  is  fy  of  1  cwt.  ? 

98.  What  is  j\  of  1  hhd.  of  wine  ? 

99.  What  is  ^\  of  1  hhd.  of  wine  1 

100.  What  is  I  of  a  yard  ? 

101.  What  is  j\  of  a  yard'? 

102.  What  is  j\  of  a  yard  ? 

103.  What  is  ^  of  a  dollar  ? 

104.  What  is  ^\  of  a  dollar? 

105.  What  is  ^  of  a  dollar  ? 

106.  What  is  ^%  of  1^.  ? 

107.  What  is  i^  of  1^.  1 

108.  What  is  ^  of  l£. 

109.  What  is  |f  of  a  gallon  of  wine  1 

110.  What  is  11  of  a  shilling  ? 

111.  What  is  ll  of  a  day  ? 

112.  What  is  -^%'j  of  a  dollar  ? 

113.  What  is  II  of  a  yard? 

114.  What  is  If  of  a  bushel  1 

115.  What  is  i-f-  of  1  lb.  Avoirdupois! 

116.  What  is  f^  of  1^.7 

117.  What  is -j%^  of  1^.  ? 

118.  What  is  if  4  on£.l 

119.  What  is  III  of  1  cwt.? 

120.  What  is  j\%^j  of  a  week  ? 

121.  What  is  III  of  1  hhd.  of  brandy  ? 

122.  What  will  Jf^  of  1  hhd.  of  wine  come  to,  at  $1.23 
per  gal.  7 

123.  What  will  ffj  of  1  cwt.  of  sugar  come  to,  at  $0.12 
per  lb.  ? 


56  ARITHMETIC.  Part  I. 

124.  What  will  4f  tons  of  iron  come  to,  at  $4.00  per  cwt.  1 

125.  What  will  7  3^  cwt.  of  sugar  come  to,  at  8  cents  per  lb.  1 

126.  What  will  8|  hhd.  of  molasses  come  to,  at  $0.48  per 
gal.] 

127.  What  will  19i|  tons  of  iron  come  to,  at  $0.05  per  lb.  1 

128.  What  will  23^  pipes  of  brandy  come  to,  at  $1.43 
per  gal,  1 

129.  At  5s.  per  bushel,  what  will  4  bu.  3  pks.  5  qts.  of 
corn  come  to  1 

130.  At  $9.00  per  cwt.,  what  will  be  the  price  of  lib.  of 
sugar  1     What  will  3  cwto  2  qrs.  7  lb.  come  to  at  that  rate  7 

131.  At  $87.00  per  cwt.,  what  cost  4  chests  of  tea,  each 
weighing  3  cwt.  3  qrs.  14  lb.  1 

132.  What  cost  18  gals.  3  qts.  of  brandy,  at  the  rate  of 
$97.00  per  hhd.  ? 

133.  Bought  a  silver  cup  weighing  9oz.  4dwt.  16  grs.  for 
3c£.  2s.  3d.  How  much  was  it  per  grain.  How  much  per 
ounce  1 

134.  Bought  a  silver  tankard  weighing  1  lb.  8  oz.  17  dwt, 
13  gr.  for  $25.00  ;  how  much  was  it  per  ounce  1 

135.  If  34  tons  9  cwt.  2  qrs.  IS  lb.  of  tallow  cost  $6,500,00, 
what  is  it  per  lb.  1    How  much  per  ton  1 

136.  A  and  B  traded  ;  A  sold  B  S\  cwt.  of  sugar,  at  12 
cents  per  lb. ;  how  much  did  it  come  to  1  In  exchange,  B 
gave  A  18  cwt.  of  flour ;  what  was  the  flour  rated  at  per  lb.  1 

137.  B  delivered  C  2  pipes  of  brandy,  at  $1.40  per  gal- 
lon, for  which  he  received  87  yards  of  cloth  ;  what  was  the 
cloth  valued  at  per  yard  ? 

138.  D  sells  E  370  yards  of  cotton  cloth  at  33  cents  per 
yard ;  for  which  he  receives  500  lb.  of  pepper  ;  what  does 
the  pepper  stand  him  in  per  lb.  1 

139.  A  merchant  bought  3  hhds.  of  brandy,  at  $1.30  per 
gallon,  and  sold  it  so  as  to  gain  |  of  the  first  cost ;  how  much 
did  he  sell  it  for  per  gallon  ? 

140.  A  merchant  bought  a  quantity  of  tobacco  for  $2-50.00, 
and  sold  it  so  as  to  gain  -^-q  of  the  first  cost ;  how  much  did 
he  sell  it  for  ? 

141.  A  merchant  bought  1  hhd  of  wine  for  $80.00  ;  how 
much  must  he  sell  it  for  to  gain  $15.00  1  How  much  will 
that  be  a  gallon  1 

142.  A  merchant  bought  500  barrels  of  flour  for  $3,000,00; 
how  much  must  he  sell  it  for  per  barrel  to  gain  $250.00 
on  the  whole  ? 


XVI.  ARITHMETIC.  57 

143.  A  merchant  bought  350  yards  of  cloth  for  $1,800,00  ; 
how  much  must  he  sell  it  for  to  gain  j\  of  the  first  cost  1 
How  much  will  that  be  a  yard  ? 

144.  A  merchant  bought  2  hhds.  of  molasses  for  835.28; 
how  much  must  he  sell  it  for  per  gal.  to  gain  §  of  the  first  cost  ? 

145.  A  merchant  bought  7  cwt.  of  coffee  for  $175.00, 
but  being  damaged  he  was  willing  to  lose  ^  of  the  first  cost. 
How  much  did  he  sell  it  for  per  lb.  ? 

146.  A  merchant  sold  7  cwt.  of  rice  for  $22.75,  to  receive 
the  money  in  6  months,  but  for  ready  money  he  agreed  to 
make  a  discount  of  j^  of  the  whole  price.  How  much  was 
the  rice  per  lb.  after  the  discount  ? 

147.  If  8  boarders  will  drink  a  cask  of  beer  in  12  days, 
how  long  would  it  last  1  boarder  ?  How  long  would  it  last 
12  boarders  ? 

148.  If  23  men  can  build  a  wall  in  32  days,  how  many 
men  would  it  take  to  do  it  in  1  day  ?  How  many  men  will 
it  take  to  do  it  in  8  days  ? 

149.  If  15  men  can  do  a  piece  of  work  in  84  days,  how- 
many  men  must  be  employed  to  perform  the  whole  in  1  day  I 
How  many  to  do  it  in  30  days  ? 

150.  If  18  men  can  perform  a  piece  of  work  in  45  days, 
how  many  days  would  it  take  1  man  to  Jo  it  1  How  long 
would  it  take  57  men  to  do  it  ? 

151.  If  25  men  can  do  a  piece  of  work  in  17  days,  in  how 
many  days  will  38  men  do  it  ? 

152.  If  a  man  perform  a  journey  in  8  days,  by  travelling 
12  hours  in  a  day,  how  many  hours  is  he  performing  it  1 
How  many  days  would  it  take  him  to  perform  it  if  he  travel- 
led only  8  hours  in  a  day  1 

153.  If  a  man,  by  working  11  hours  in  a  day,  perform  a 
piece  of  work  in  24  days,  how  many  days  will  it  take  him  to 
do  it  if  he  works  13  hours  in  a  day  ? 

154.  If  I  can  have  5  cwt.  carried  138  miles  for  11  dol- 
lars, how  far  can  I  have  25  cwt.  carried  for  the  same  money  1 

155.  Suppose  a  man  agrees  to  pay  a  debt  with  wheat,  and 
that  it  will  take  43  bushels  to  pay  it,  when  wheat  is  7  shil- 
lings per  bushel ;  how  much  will  it  take  when  wheat  is  9 
shillings  per  bushel  ? 

156.  If  11  men  can  do  a  piece  of  work  in  14  days,  when 
the  days  are  15  hours  long,  how  many  men  would  it  take  to 
do  it  in  the  same  number  of  days,  when  the  days  are  11  hours 
long  ? 


58  ARITHMETIC.  Part  1. 

157.  If  5  men  can  do  a  piece  of  work  in  5  months  by 
working  7  hours  in  a  day,  in  how  many  months  will  they  do 
it,  if  they  work  10  hours  in  a  day  ? 

158.  Two  men,  A  and  B,  traded  in  company  ;  A  furnish- 
ed f  of  the  stock  and  B  -^ ;  they  gained  $864.00  ;  what  was 
each  one's  share  of  the  gain  1 

159.  Three  men,  A,  B,  and  C,  traded  in  company  ;  A 
furnished  ^  of  the  capital ;  B  i|,  and  C  the  rest.  They 
gained  $8,453,67 ;  what  was  each  one's  share  of  the  divi- 
dend 1 

160.  Two  men,  B  and  C,  bought  a  barrel  of  flour  to- 
gether. B  paid  5  dollars  and  C  3  dollars ;  what  part  of  the 
whole  price  did  each  pay  1  What  part  of  the  flour  ought 
each  to  have  ? 

161.  Two  men,  C  and  D,  bought  a  hogshead  of  wine  ;  C 
paid  $47.00,  and  D  53.00 ;  how  many  dollars  did  they  both 
pay  ?  What  part  of  the  whole  price  did  each  pay  1  How 
many  gallons  of  the  wine  ought  each  to  have  1 

162.  Three  men,  C,  D,  and  E,  traded  in  company  ;  C  put 
m  $850.00;  D,  942.00;  and  E,  $1,187.00;  how  many 
dollars  did  they  all  put  in  1  What  part  of  the  whole  did 
each  put  in  ?  They  gained  $1,353.18 ;  what  was  each 
man's  share  of  the  gain  ? 

163.  Five  men.  A,  B,  C,  D,  and  E,  freighted  a  vessel : 
A  put  on  board  goods  to  the  amount  of  $4,000.00 ;  B, 
$15,000.00  ;  C,  $11,000.00 ;  D,  $7,500.00 ;  and  E,  $850.00. 
During  a  storm  the  captain  was  obliged  to  throw  overboard 
goods,  to  the  amount  of  $13,400.00;  what  was  each  man's 
share  of  the  loss  1 

164.  Three  men  bought  a  lottery  ticket  for  $20.00  ;  of 
which  F  paid  $4.37  ;  G  $8.53  ;  and  H,  the  rest.  They 
drew  a  prize  of  $15,000.00  ;  what  was  the  share  of  each  1 

165.  Three  men  hired  a  pasture  for  $42.00  ;  the  first  put 
in  4  horses  ;  the  second,  6  ;  and  the  third,  8.  What  ought 
each  to  pay  ? 

166.  A  man  failing  in  trade,  owes  to  A  $350.00  ;  to  B 
$783.00  ;  to  C  $1,780.00  ;  to  D  $2,875.00  ;  and  he  has  only 
$2,973.00  in  property,  which  he  agrees  to  divide  among  his 
creditors  in  proportion  to  the  several  debts.  What  will  each 
receive  1 

167.  What  is  -i-i-Jj  of  378,648 1 

168.  What  is  iff  I  of  87? 

169.  What  is  rfls  of3? 


XVII.  ARITHMETIC.  59 

170.  What  is  ^6_%  of  47? 

171.  Multiply  14"!  by  7. 

172.  What  is  ,5 14  of  7? 

173.  Multiply  973  by  ^§1. 

174.  Multiply  ffl-  by  973. 

175.  Multiply  471  by  j\f^. 

176.  Multiply  2V-  by  471. 

177.  Multiply  /^oV  by  138. 

178.  Multiply  138  by  j^^. 

179.  Multiply  -j-fl-j  by  9o0. 
ISO.  Multiply  950  by  yff^. 

XVII.  1.  If  2  lb.  of  figs  cost  |  of  a  dollar,  what  is  that  a 
pound  ? 

2.  If  2  bushels  of  potatoes  cost  |  of  a  dollar,  what  is  that 
a  bushel  1  What  would  be  the  price  of  8  bushels  at  that 
rate  ? 

3.  If  I  of  a  barrel  of  flour  were  to  be  divided  equally 
among  3  men,  how  much  would  each  have  ? 

4.  If  3  horses  consume  j\  of  a  ton  of  hay  in  1  month,  how 
much  will  1  horse  consume  1  How  much  would  1 1  horses 
consume  in  the  same  time  1 

5.*  If  3  lb.  of  beef  cost  }f  of  a  dollar,  what  would  a  quar- 
ter of  beef,  weighing  136  lb.,  cost  at  that  rate  ? 

6.  If  2  yds.  of  cloth  cost  S|  dollars,  what  will  7  yards  tost 
at  that  rate  ? 

7.  If  4  bushels  of  wheat  cost  32|  shillings,  what  will  17 
bushels  cost  ? 

8.  If  3  sheep  are  worth  23|  bushels  of  wheat,  how  many 
bushels  is  1  sheep  worth  ?  How  many  bushels  are  50  sheep 
worth  at  that  rate  1 

Note.  Reduce  23|  to  fifths,  or  divide  as  far  as  you  can, 
and  then  reduce  the  remainder  to  fifths,  and  take  -^  of 
them. 

9.  If  7  calves  are  worth  59-i-  bushels  of  corn,  how  many 
bushels  are  15  calves  worth  at  that  rate  ? 

10.  A  man  laboured  15  days  for  20f  dollars  ;  how  much 
would  he  earn  in  3  months  at  that  rate,  allowing  26  working 
days  to  the  month  ? 

1 1.  A  man  travelled  88yL.  miles  in  17  hours  ;  how  far  did 
he  travel  in  an  hour  1 

*  See  First  Lessons,  Sect.  XFV. 


60  ARITHMETIC.  Part  1. 

12.  A  man  travelled  4764  miles  in  8  days ;  how  far  did 
he  travel  each  day,  supposing  he  travelled  the  same  number 
of  miles  each  day  1 

13.  Divide  77y\  bushels  of  corn  equally  among  15  men. 

14.  If  23  yards  of  cloth  cost  175-^  dollars,  what  is  that  a 
yard  1 

15.  If  35  lb.  of  raisins  cost  Sj^/q-  dollars,  what  will  2  cwt. 
cost  at  that  rate  1 

16.  A  man  divided  |^  of  a  water-melon  equally  between  2 
boys ;  how  much  did  he  give  them  apiece  1 

17.  Suppose  you  had  :j  of  a  pine  apple  and  should  divide 
it  into  two  equal  parts  ;  what  part  of  the  whole  apple  would 
each  part  be  ? 

18.  If  you  dividef  of  a  bushel  of  corn  equally  between  2 
men,  how  much  would  you  give  them  apiece  ? 

19.  What  is  i  of -3? 

20.  If  you  divide  ^  of  a  bushel  of  grain  between  two  men, 
how  much  would  you  give  them  apiece  ? 

Note.  Cut  the  third  into  two  parts.  What  will  the  parts 
be? 

21.  What  is  i  of  i? 

22.  If  you  divide  i  of  a  barrel  of  flour  equally  bniween 
two  men,  how  much  will  you  give  them  apiece  1 

23.  What  is  i- of  j? 

24.  A  man  having  |  of  a  barrel  of  flour  divided  it  equally 
among  4  men  ;  how  much  did  he  give  them  apiece  1 

25.  What  is  i  off? 

26.  If  3  lb.  of  sugar  costf^  of  a  dollar,  what  is  it  a  pound  ? 

27.  What  is  I  off? 

28.  If  5  lb.  of  rice  cost  |  of  a  dollar,  what  is  that  a  pound  ? 

29.  If  3  lb.  of  raisins  cost  -^  of  a  dollar,  what  is  that  a 


pound  ?  What  will  2  lb.  cost  at  that  rate  ?  What  7  lb.  ? 

30.  What  is  i  of  i  ?     What  is  f  of  i  ?     What  is  ^  of  i  ? 

31.  If  7  lb.  of  sugar  cost  f  of  a  dollar,  what  is  it  a  pound  ? 
What  will  5  lb.  cost  at  that  rate  ?     What  would  15  lb.  cost  1 

32.  What  is  4  of  I  ?  What  is  f  of  f  ?  What  is  y  of  f  ? 

33.  During  a  storm,  a  master  of  a  vessel  v/as  obliged  to 
throw  overboard  -^-^  of  the  whole  cargo.  What  part  of  the 
whole  loss  must  a  man  sustain  who  owned  ^  of  the  cargo  ? 

34.  A  man  owned  2  j  of  the  capital  of  a  cotton  manufac- 
tory, and  sold  y4_.  of  his  share.  What  part  of  the  whole  cap- 
ital did  he  sell  ?     What  part  did  he  then  own  '' 


XVII.  ARITHMETIC.  61 

35.  If  3  bushels  of  wheat  cost  5-^  dollars,   what  is  it  a 
bushel  ?     What  will  2  bushels  cost  dt  that  rate  ? 
86.  What  is  ^  of  5^  ?    What  is  f  of  54:  ? 

37.  If  4  dollars  will  buy  5|  bushels  of  rye,  how  much  will 
one  dollar  buy  ?     How  much  will  3  dollars  buy  ? 

38.  What  is  i  of  55.  ?     What  is  |  of  5|  ? 

39.  If  17  ban-els  of  flour  cost  $107f,  what  will  23  barrels 
cost? 

40.  What  is  f  f  of  107f  ? 

41.  If  12  cwt.  of  sugar  cost  $137|,  what  is  the  price  of 
1  qr.  1     What  of  1  lb.  ? 

42.  At  4  dollars  for  3^  gallons  of  wine,  how  much  may  be 
bought  for  07i  dollars  ? 

Note.     Find  how  much  |  a  dollar  will  buy. 

43.  If  3  cords  of  wood  cost  20  dollars,  what  will  7^  cords 
cost? 

44.  If  19  yards  of  cloth  cost  155  dollars,  what  will  be  the 
price  of  1-|-  yards? 

45.  If  18  lb.  of  raisins  cost  2f  dollars,  what  is  that  per 
lb.  ?    What  would  be  the  price  of  5|  lb.  at  that  rate  ? 

46.  If  11  lb.  of  butter  cost  2^^^  dollars,  what  will  18f  lb. 
cost? 

47.  If  7  gallons  of  vinegar  cost  f  of  a  dollar,  what  will 
27  |-  gallons  cost  ? 

48.  If  1  lb.  of  sugar  cost  |i  of  a  dollar,  what  will  17|  lb. 
cost? 

49.  If  a  yard  of  cloth  cost  7j\  dollars,  what  will  |  of  a 
yard  cost  ? 

50.  At  2*j  of  a  dollar  a  yard,  what  will  ^  of  a  yard  of 
cloth  cost  ? 

51.  At  31  shillings  a  yard,  what  will  7|  yards  of  riband 
cost? 

52.  At  3  dollars  a  barrel,  what  part  of  a  barrel  of  cider 
may  be  bought  for  ^  of  a  dollar  ? 

53.  At  4  dollars  a  yard,  what  part  of  a  yard  of  cloth  may 
be  bought  for  i  of  a  dollar  ? 

54.  At  2  dollars  a  yard,  how  much  cloth  may  be  bought 
for  .51  dollars  ? 

55.  At  2  dollars  a  gallon,  how  much  brandy  may  be  bought 
for  7f  dollars  ? 

56.  At  3  shilliags  a  quart,  how  many  quarts  of  wine  may 
be  bought  for  17|  shillings  ? 

6 


62  ARITHMETIC.  Part  1. 

57.  At  6  dollars  a  barrel,  how  many  barrels  of  flour  may 
be  bought  for  45y3-  dollars  ? 

58.  If  1  cwt.  of  iron  cost  4|  dollars,  what  will  5f-  cwt. 
cost  1 

59.  A  man  failing  in  trade  can  pay  only  |  of  a  dollar  on 
each  dollar,  how  much  can  he  pay  on  7i  dollars  1  How 
much  on  23f  dollars  ? 

60.  A  man  failing  in  trade  is  able  to  pay  only  if  of  a 
pound  on  a  pound,  how  much  can  he  pay  on  \.1£.  15s.  1 

61.  A  man  failing  in  trade  is  able  to  pay  only  17s.  on  a 
pound,  what  part  of  each  pound  will  he  pay  1  How  much 
will  he  pay  on  a  debt  of  147^.  14s.  ] 

62.  What  is  i  of  11? 

63.  Divide  f  ^  by  6. 

64.  Multiply  Vo^5  by  \, 

65.  What  is  J^^  of  f  1 

66.  Multiply  |f  by  ^j, 

67.  Divide  f  ^  by  25". 

68.  Divide  lo^^  by  8. 

69.  Multiply  15|f  by  ^. 

70.  What  is  ^Vt  of  \lj\  1 

71.  Multiply  l3f  by  xV 

72.  Multiply  135v«T-  by  24|. 

73.  Multiply  l,647f  by  17f|. 

74.  How  many  times  is  3  contained  in  14f  1 

75.  How  many  times  is  9  contained  in  47y*j  1 

76.  How  many  times  is  17  contained  in  2534  f  *? 

77.  What  part  of  2  is  f? 

78.  AVhatpartof7isy4_? 

79.  What  part  of  19  is  |I  ? 

80.  What  part  of  123  is  y^j  "? 

81.  What  part  of  8  is  7|? 

82.  What  part  of  19  is  14|  1 

83.  What  part  of  82  is  19f J? 

84.  What  part  of  125  is  47/^  1 


XVIII.     1.  If  1  lb.  butter  cost  ^  of  a  dollar,  how  much 
fvill  2  lb.  cost  ?     What  will  4  lb.  cost  ? 

2.  At  i  of  a  dollar  per  lb.,  what  will  2  lb.  of  raisins  cost  t 
What  will  3  lb.  cost  1     What  will  6  lb.  cost  1 

3.  If  1  man  will  consume  f  of  a  bushel  o/  corn  in  a  week, 
how  much  will  2  men  consume  in  the  same  time  1     How 


JCVIII.  ARITHMETIC.  63 

much  will  4  men  consume  1     How  much  will  8  men  con- 
sume? 

4.  If  a  horse  will  consume  |  of  a  bushel  of  oats  in  a  day, 
how  much  will  he  consume  in  3  days  ?  How  much  in  9  days  1 

5.  If  1  man  can  do  j'^  ^f  a  piece  of  work  in  a  day,  how 
much  of  it  can  2  men  do  in  the  same  time  1  How  much 
of  it  can  3  men  do  ?  How  much  of  it  can  4  men  do  1  How 
much  of  it  can  6  men  do  1    How  much  of  it  can  12  men  do  1 

6.  If  a  man  drink  j\  of  a  barrel  of  cider  in  a  week,  how 
much  would  he  drink  in  2  weeks  ?  How  much  would  5 
men  drink  in  a  week  at  that  rate  ?  How  much  would  8  men 
drink  in  a  week  ?  How  much  would  20  men  drink  in  a 
week  ?  How  much  would  40  men  drink  in  a  week  1 

7.  If  a  horse  consume  2f  bushels  of  oats  in  a  week,  how 
much  would  he  consume  in  4  weeks  1  How  much  in  8 
weeks  ? 

8.  At  72^^  dollars  a  barrel,  what  cost  5  barrels  of  flour  1 

9.  If  a  horse  will  eat  ^-^  of  a  ton  of  hay  in  a  month,  how 
much  will  2  horses  eat  ?    How  much  will  8  horses  eat  1 

10.  If  it  take  l||  yard  of  cloth  to  make  a  coat,  how  much 
will  it  take  to  make  8  coats  ?    How  much  to  make  24  coats  1 

11.  If  a  barrel  of  cider  cost  Oj^g-  dollars,  what  will  10 
barrels  cost  ?     What  will  25  barrels  cost  1 

12.  Multiply  ^  by  5. 

13.  Multiply  II  by  8. 

14.  Multiply  -1/-J  by  25. 

15.  Multiply  ^Ve  by  8. 

16.  Multiply  Uj-  by  9. 

17.  Multiply  Ul  by  4. 

18.  Multiply  /oVo  by  100. 

19.  Multiply  43||-  by  28. 

20.  Multiply  137 jVy  by  3. 

21.  Multiply  I  by  .8. 

Note.  8  times  1=1  ;  8  times  J  is  7  times  as  much,  that 
is,  7.     Perform  the  following  examples  in  a  similar  manner. 

22.  How  much  is  7  times  ^  ? 

23.  How  much  is  19  times  \^  1 

24.  How  much  is  23  times  i|  1 

25.  Multiply  7|  by  5. 

26.  Multiply  19^  by  17. 

27.  Multiply  123^  by  9.  - 

28.  Multiply  43^11  by  327. 


64  ARITHMETIC.  Part  L 

29.  Multiply  9f //^  by  126S. 

30.  Multiply  14yf  §0  by  1000. 

XIX.  1.*  A  merchant  bought  4  pieces  of  cloth,  the  first 
contained  18|  yards,  the  second  27^  yards,  the  third  23f 
yards,  and  the  fourth  25f  yards.  How  many  yards  in  the 
whole  ? 

2.  A  gentleman  hired  2  men  and  a  boy  for  1  week.  One 
man  was  to  receive  5|  dollars,  the  other  7{r,  and  the  boy  3|. 
How  much  did  he  pay  the  whole  1 

3.  A  gentleman  hired  three  men  for  1  month.  To  the 
first  he  paid  '^dj-o  bushels  of  corn  ;  to  the  second,  28yo-  bush 
els,  and  to  the  third,  33y^o  bushels.  How  many  bushels  did 
it  take  to  pay  them  ? 

4.  A  man  had  2}  bushels  of  corn  in  one  sack,  and  2|-  in 
another  ;  how  many  bushels  had  he  in  both  ? 

5.  If  it  takes  1^  yard  of  cloth  to  make  a  coat,  and  f  of  a 
yard  to  make  a  pair  of  pantaloons,  how  much  will  it  take  to 
make  both  1 

6.  A  man  bought  2  boxes  of  butter  ;  one  had  7J  lb.  in  it, 
and  the  other  lOf  lb.     How  many  pounds  in  both  1 

7.  A  boy  having  a  pine  apple,  gave  \  of  it  to  one  sister,  \ 
to  another,  and  \  to  his  brother,  and  kept  the  rest  himself. 
How  much  did  he  keep  himself? 

8.  A  man  bought  3  sheep ;  for  the  first  he  gave  6J  dol 
lars  ;  for  the  second,  8f  ;  and  for  the  third,  9|.  How  many 
dollars  did  he  give  for  the  whole  ? 

9.  How  many  cvvt.  of  cotton  in  four  bags  containing  as 
follows  ;  the  first  4|  cwt. ;  the  second,  5f  cwt. ;  the  third 
4j^^  cwt ;  and  the  fourth  H-^-^  cwt.  ? 

10.  A  merchant  bought  a  piece  of  cloth  containing  23 
yards,  and  sold  l^r  yards  of  it ;  how  many  yards  had  he  left  ? 

11.  A  gentleman  paid  a  man  and  a  boy  for  2  months'  la- 
oour  with  corn  ;  to  the  man  he  gave  26f-  bushels,  and  to  the 
boy  he  gave  18f  bushels.  How  many  bushels  did  it  take  to 
pay  both  ? 

12.  Bought  8|-  cwt.  of  sugar  at  one  time,  and  5f  cwt.  at 
another ;  how  much  in  the  whole  ? 

13.  Bought  y  of  a  ton  of  iron  at  one  time,  and  ^-  of  a  ton 
at  another  ;  how  much  in  the  whole  ? 

14.  There  is  a  pole  standing  so  that  -|  of  it  is  in  the  mud, 

*  See  First  Leasons,  Sect.  XIII 


XX.  ARITHMETIC.  65 

I  of  it  in  the  water,  and  the  rest  above  the  water ;  how  much 
of  it  is  above  the  water  ? 

15.  A  merchant  bought  14|i  cwt.  of  sugar,  and  sold  8^, 
cwt.  ;  how  many  lb.  had  he  left  1 

Note,  Reduce  all  fractions  to  their  lowest  terms,  after 
the  work  is  completed,  or  before  if  convenient.  In  the  above 
example  -/y  might  be  reduced,  but  it  would  not  be  convenient 
because  it  now  has  a  common  denominator  with  \}.  The 
answer  may  be  reduced  to  lower  terms. 

16.  Out  of  a  barrel  of  cider  there  had  leaked  7f  gallons 
how  many  gallons  were  there  left  ? 

17.  A  man  bought  2  loads  of  hay,  one  contained  IT-J 
cwt.  and  the  other  23^'V  ^^^*-  ^^^^  many  cwt.  in  both  ? 

18.  A  man  had  43^^  cwt.  of  hay,  and  in  3  weeks  his  horse 
ate  5-i^  cwt.  of  it ;  how  much  had  he  left  7 

19.  Two  boys  talking  of  their  ages,  one  said  he  was  9| 
years  old ;  the  other  said  he  was  4^*y  years  older.  What 
was  the  age  of  the  second  ? 

20.  A  lady  being  asked  her  age,  said  that  her  husband 
was  37|-  years  old,  and  she  was  not  so  old  as  her  husband  by 
3_9_  years.     What  was  her  age  1 

21.  A  lady  being  asked  how  much  older  her  husband  was 
than  herself,  answered,  that  she  could  not  tell  exactly  ;  but 
when  she  was  married  her  husband  was  28y\  years  old,  and 
she  was  22-|.     What  was  the  difference  of  their  ages  1 

22.  Add  together  |-  and  -^3 . 

23.  Add  together  |,  f ,  and  J. 

24.  Add  together  y\  and  -^\. 

25.  Add  together  13^  and  172^- 

26.  Add  together  137f ,  26^,  and  243f . 

27.  What  is  the  difference  between  f  and  1 1 

28.  What  is  the  difference  between  -^  and  |f  1 

29.  What  is  the  difference  between  13-,^  and  8/1  1 

30.  What  is  the  difference  between  137|  and  98f  1 

31.  Subtract  38-i^^  from  53^. 

32.  Subtract  284^  from  S13|4. 

XX.  1.  A  man  bought  15  cows  for  $345.  What  was  the 
average  price  1 

Note.     Find  the  price  of  3  cows,  and  then  of  1  cow. 

2.  A  merchant  bought  16  yards  of  cloth  for  $84.64  ;  what 
was  it  a  yard  ? 


66  ARITHMETIC.  Part.  L 

3.  A  merchant  bought  18  barrels  of  flour  for  $  114.66, 
and  sold  it  so  as  to  gain  $1.00  a  bbl.  How  much  did  he  sell 
it  for  per  barrel  ] 

4.  21  men  are  to  share  equally  a  prize  of  8,530  dollars,  how 
much  will  they  have  apiece  ? 

5.  A  merchant  sold  a  hogshead  of  wine  for  113  dollars 
How  much  was  it  a  gallon  ? 

6.  A  ship's  crew  of  30  men  are  to  share  a  prize  of  847 
dollars  ;  how  much  will  they  receive  apiece  1 

7.  A  man  has  1.857  lb.  of  tobacco,  which  he  wishes  to 
put  into  42  boxes,  an  equal  quantity  in  each  box.  How  much 
must  he  put  into  each  box  ? 

8.  In  4,847  gallons  of  wine,  how  many  hogsheads  1 

9.  At  $48.00  a  ba/rel,  how  many  barrels  of  brandy  may 
be  bought  for  $687.43 1 

10.  At  $90  dollars  a  ton,  how  many  tons  of  iron  may  be 
bought  for  2,486  dollars  ? 

11.  If  23.000cwt.ofiron  cost  $92,368.75,  how  much  is  it 
per  lb.  ? 

12.  Divide  784  by  28. 

13.  Divide  1,008  by  36. 

14.  Divide  1,728  by  72. 

15.  Divide  2,352  by  56. 

16.  Divide  183  by  15. 

17.  Divide  487  by  18. 
IS.  Divide  1,243  by  25. 

19.  Divide  37,864  by  6S. 

20.  Divide  19,743  by  112. 

21.  Divide  4,383  by  30. 

22.  Divide  6,487  by  50. 

23.  Divide  1,673  by  400. 

24.  Divide  13,748  by  7,000. 

25.  Divide  100,780  by  250. 

26.  Divide  406,013  by  4,700. 

27.  Divide  3,000,406  by  306,000. 

28.  Divide  450,387  by  36,000. 

29.  Divide  78,407,300  by  42,000. 

30.  Divide  15,008,406  by  480,000« 

XXI.  1.  Find  the  divisors  of  each  of  the  following  nuiJi- 
bers,  15,  18,  20,  21,  24,  28,  42,  48,  64,  72,  88,  98. 

2.  Find  the  divisors  of  each  of  the  following  numbers^ 
108,  112,  114,  120,  387,  432,  846,  936. 


XXII.  ARITHMETIC.  67 

3.  Find  the  divisors  of  each  of  the  following  numbers, 
8000,  4,053,  1,8G4,  2,480,  24,876,  103,284,  and  7,328,472, 

4.  Find  the  common  divisors  of  8  and  24. 

5.  Find  the  common  divisors  of  IG  and  36 

6.  Find  the  common  divisors  of  18  and  42 

7.  Find  the  common  divisors  of  21  and  56. 

8.  Find  the  common  divisors  of  .56  and  264. 

9.  Find  the  common  divisors  of  123  and  642. 

10.  Find  the  common  divisors  of  32,  96,  and  1,432. 

11.  Find  the  common  divisors  of  7,362,  and  2,484. 

12.  Find  the  common  divisors  of  73,647,  84,177,  and 
i,684. 

13.  Reduce  i|-  to  its  lowest  terms. 

14.  Reduce  /(^^  to  its  lowest  terms. 

15.  Reduce  Jf ^  to  its  lowest  terms. 

16.  Reduce  -^%  to  its  lowest  terms. 

17.  Reduce  ^j%\  to  its  lowest  terms. 

18.  Reduce  j^-^^\  to  its  lowest  terms. 

19.  Reduce  ^f-^-fo  to  its  lowest  terms. 

XXII.     1.  Reduce  J  and  f  to  the  least  common  denomr- 
aator. 

2.  Reduce  J  and  ^  to  the  least  common  denominator. 

3.  Reduce  |-  and  |  to  the  least  common  denominator. 

4.  Reduce  f  and  -f-^  to  the  least  common  denominator. 

5.  Reduce  y%  and  -/^  to  the  least  common  denominator. 

6.  Find  the  least  common  multiple  of  8  and  12. 

7.  Find  the  least  common  multiple  of  8  and  14. 

8.  Find  the  least  common  multiple  of  9  and  15. 

9.  Find  the  least  common  multiple  of  15  and  18. 

10.  Find  the  least  common  mulnple  of  10,  14,  and  15. 

11.  Find  the  least  common  multiple  of  15,  24,  and  35. 

12.  Find  the  least  common  multiple  of  30,  4S,  and  56. 

13.  Find  the  least  common   multiple  of  32,  72,  and  120. 

14.  Find  the  least  common  multiple  of  42,  60,  and  125. 

15.  Find  the  least  common  multiple  of  250,  180,  and  540- 

16.  Reduce  -j\  and  /^  to  the  least  common  denominator. 

17.  Reduce  /y  and  ^  to   the  least  common  denomina- 
or. 

18.  Reduce  f  g ,  2V'  and  ^,  to  the  least  common  denomi- 
hator. 

19.  Reduce  |,  f ,  y\,  and  /y,  to  the  least  common  denomi- 
nator. 


68  ARITHMETIC.  Part  J. 

20.  Reduce  2^,  g^,  and  -fj  to  the  least  common  denomi- 
nator. 

21.  Reduce  ^^  and  -g^  to  the  least  common  denomina- 
tor. 

22.  Reduce  -^^  and  2T0V0  ^^  ^^^  ^^^^  common  denomi- 
nator. 

23.  Reduce  ^y^  and  ^Irw  ^-^  ^he  least  common  denomi- 
nator. 

24.  Reduce  -^l^  and  x§wo  ^^  the  least  common  de- 
nominator. 

XXIII.  1.*  At  -^  of  a  dollar  a  bushel,  how  many  bushels 
of  potatoes  may  be  bought  for  5  dollars  1  How  many  at  | 
of  a  dollar  a  bushel  ? 

2.  At  4  of  a  shilling  apiece,  how  many  peaches  may  be 
bought  for  a  dollar  ?     How  many  at  f  of  a  shilling  apiece  ? 

3.  A  gentleman  distributed  6  bushels  of  corn  among  some 
labourers,  giving  them  ^  of  a  bushel  apiece ;  how  many  did 
he  give  it  to  1  How  many  would  he  have  given  it  to,  if  he 
had  given  ^  of  a  bushel  apiece  1 

4.  If  it  takes  f  of  a  bushel  of  rye  to  sow  1  acre,  how  many 
acres  will  15  bushels  sow  7 

5.  A  merchant  had  47  cwt.  of  tobacco  which  he  wished 
to  put  into  boxes,  containing  ^V  ^^t.  each.  How  many  boxes 
must  he  get  ? 

6.  A  gentleman  has  a  hogshead  of  wine  which  he  wishes 
to  put  into  bottles,  containing  -^-^  of  a  gallon  each.  How 
many  bottles  will  it  take  1 

7.  If  3^0-  of  a  barrel  of  cider  will  last  a  family  1  week,  how 
many  weeks  will  7  barrels  last  ? 

8.  If /j  ^f  ^  bushel  of  grain  is  sufficient  for  a  family  of 
two  persons  1  day,  how  many  days  would  16  bushels  last? 
How  many  persons  would  16  bushels  last  1  day  1 

9.  If  a  labourer  drink  ||  of  a  gallon  of  cider  in  a  day,  one 
day  with  another,  how  long  will  it  take  him  to  drink  a  hogs- 
head 1 

10.  If  an  axe-maker  put  y^  of  a  lb.  of  steel  into  an  axe, 
how  many  axes  would  I  cwt.  of  steel  be  sufficient  for  1 

11.  If  it  take  1^  bushel  of  oats  to  sow  an  acre,  how  many 
acres  will  18  bushels  sow  ? 

12.  If  it  take  1^  bushel  of  wheat  to  sow  an  acre,  how 
many  acres  will  23  bushels  sow  ? 

*  See  First  Lessons,  Sect.  XV 


XXIII.  ARITHMETIC.  GO 

13.  At  1|  dollar  a  bushel,  how  much  wheat  may  be 
bought  for  20  dollars  ? 

14.  At  th}  dollars  a  barrel,  how  many  barrels  of  cider  may 
be  bought  for  40  dollars  ? 

15.  At  the  rate  of  15f  bushels  to  the  acre,  how  many 
acres  will  it  take  to  produce  75  bushels  of  rye  ? 

16.  At  4|  dollars  per  cwt.,  how  many  tons  of  iron  can  I 
buy  for  $150? 

17.  At  llf  cents  per  lb.,  how  much  steel  can  I  buy  for 
$50.00  ? 

18.  If  a  man  can  perform  a  journey  in  580  hours,  how 
many  days  will  it  take  him  to  perform  it  if  he  travel  9j% 
hours  in  a  day  1 

19.  How  many  coats  may  be  made  of  187  yards  of  cloth 
if  3y\  yards  make  1  coat  ? 

20.  In  43  yards  how  many  rods  1 

21.  In  87  yards  how  many  rods  ? 

22.  In  853  feet  how  many  rods  1 

23.  In  2,473  feet  how  many  furlongs  1 

24.  In  43^872  feet  how  many  miles? 

25.  If  1  bushel  of  apples  cost  :^  of  a  dollar,  how  many 
bu-shels  may  be  bought  for  f  of  a  dollar  ? 

26.  At  |-  of  a  dollar  a  dozen,  how  many  dozen  of  lemons 
may  be  bought  for  f  of  a  dollar  ?  How  many  dozen  for  If 
dollar  ? 

27.  At  |-  of  a  dollar  a  dozen,  how  many  dozen  of  oranges 
may  be  bought  for  y  of  a  dollar  ?    How  many  for  2|  dollars  ? 

28.  At  I  of  a  dollar  a  bushel,  how  many  bushels  of  ap- 
ples may  be  bought  for  |-  of  a  dollar  ?  How  many  for  5^ 
dollars  ? 

29.  At  ^  of  a  dollar  per  lb.,  how  many  pounds  of  figs  may 
be  bought  for  f  of  a  dollar  ?  How  many  pounds  for  1^ 
dollar  ? 

30.  At  I  of  a  dollar  a  bushel,  how  many  bushels  of  apples 
may  be  bought  for  1^  dollar  ? 

31.  If  I  of  a  chaldron  of  coal  will  supply  a  fire  1  week, 
how  many  weeks  will  J  of  a  chaldron  supply  it  ? 

32.  If  1  lb.  of  sugar  cost  |  of  a  dollar,  how  many  pounds 
may  be  bought  for  J  of  a  dollar  ?  How  many  pounds  for  1^ 
dollar  ? 

33.  At  I  of  a  dollar  per  bushel,  how  many  bushels  of  ap^- 
pies  may  be  bought  for  y  of  a  dollar  ?  How  many  at  f  of  a 
dollar  per  bushel  ? 


70  ARITHMETIC.  Part  I. 

34.  At  4-  of  a  dollar  per  bushel,  how  many  bushels  of  po- 
tatoes may  be  bought  for  |  of  a  dollar  1  How  many  at  f-  of 
a  dollar  per  bushel  ? 

35.  At  I  of  a  dollar  a  bushel,  how  much  corn  may  be 
bought  for  \  of  a  dollar  ?     How  much  for  |  of  a  dollar  7 

36.  At  f  of  a  dollar  per  bushel,  how  much  rye  may  be 
bought  for  I  of  a  dollar  ?     How  much  for  f  of  a  dollar  ? 

37.  At  -i-  of  a  shilling  apiece,  how  many  eggs  may  be 
bought  for  f  of  a  dollar  1 

38.  If  it  take  Jj  of  a  pound  of  flour  to  make  a  penny-loaf, 
how  many  penny-loaves  may  be  made  of  ^  of  a  pound  ] 

39.  If  a  four-penny  loaf  weigh  y\  of  a  pound,  how  many 
will  weigh  J  of  a  pound  1 

40.  If  a  two-penny  loaf  weigh  -f-^  of  a  pound,  how  many 
will  weigh  1|  lb  ?     How  many  will  weigh  7-|  lb.  ? 

41.  If  a  six-penny  loaf  weigh  -^^  of  a  pound,  how  many 
six-penny  loaves  will  weigh  |-  of  a  pound  1  How  many  will 
weigh  4|  lb  1 

42.  Iff  of  a  pound  of  fur  is  sufficient  to  make  a  hat,  how 
many  hats  may  be  made  of  ^-^  lb.  of  fur  1 

43.  If  10  oz.  of  fur  is  sufficient  to  make  a  hat,  how  many 
hats  may  be  made  of  4  lb.  7  oz.  of  fur  1 

44.  If  1  bushel  of  apples  cost  ff  of  a  dollar,  how  many 
bushels  may  be  bought  for  3|-  dollars  ? 

45.  If  a  bushel  of  apples  cost  2s.  5d.  how  many  bushels 
may  be  bought  for  3  dollars  and  5  shillings  1 

40.  If  1|,  that  is,  |  of  a  yard  of  cloth  will  make  a  coat, 
how  many  coats  may  be  made  from  a  piece  containing  43|- 
yards  ? 

47.  If  2i  bushels  of  oats  will  keep  a  horse  1  week,  how 
long  will  ISf  bushels  keep  him  1 

48.  If  4-^  yards  of  cloth  will  make  a  suit  of  clothes,  how 
many  suits  will  87|  yards  make  1 

49.  If  a  man  can  build  A^-^  rods  of  wall  in  a  day,  how  many 
days  will  it  take  him  to  build  84^^  rods  I 

50.  If  ff  of  a  ton  of  hay  will  keep  a  cow  through  the  win 
ter,  how  many  cows  will  23.^^^  tons  keep  at  the  same  rate  1 

51.  At  9|^  dollars  a  chaldron,  how  many  chaldrons  of 
coal  may  be  bought  for  37|-  dollars  ? 

52.  At  14 /j  dollars  per  cwt.,  how  many  cwt.  of  yellow 
ochre  may  be  bought  for  243^-^  dollars  ? 

53.  At  252  Y  dollars  a  cask,  how  many  casks  of  claret  wine 
may  be  boughl  for  3S7-/3-  dollars  ? 


XXIV.  ARITHMETIC.  71 

54.  At  95if  dollars  a  ton,  how  mirch  iron  may  be  bought 
for  2,9561  dollars  ? 

55.  How  many  times  is  /y  contained  in  17  ? 

56.  How  many  times  is  |f  contained  in  83  ? 

57.  How  many  times  is  19|4  contained  in  253  ? 

•     58.  How  many  times  is  42^%*^  contained  in  1,677  I 

59.  How  many  times  is  |-  contained  in  14^  ? 

60.  How  many  times  is  -^j  contained  in  37f  ? 

61.  How  many  times  is  3f  contained  in  24|-  ? 

62.  How  many  times  is  15^1^  contained  in  I03if  ? 

63.  How  many  times  is  27^  contained  in  1 ,605|-  ? 

64.  At  3  dollars  a  barrel,  what  part  of  a  barrel  of  cider 
may  be  bought  for  |  of  a  dollar  ? 

65.  At  7  dollars  a  barrel,  what  part  of  a  barrel  of  flour 
may  be  bought  for  |  of  a  dollar  ?  What  part  for  |  of  a  dol- 
lar ? 

66.  At  11|  dollars  per  cwt.,  what  part  of  1  cwt,  of  sugar 
may  be  bought  for  -g-  of  a  dollar  ?  What  part  of  1  cwt.  may 
be  bought  for  |  of  a  dollar  ?     What  part  for  3|  dollars  ? 

67.  At  93|  dollars  per  ton,  what  part  of  a  ton  of  iron  may 
be  bought  for  25 1-  dollars  ? 

68.  When  corn  is  |-  of  a  dollar  a  bushel,  what  part  of  a 
bushel  may  be  bought  for  |  of  a  dollar  ? 

69.  Two  men  bought  a  barrel  of  flour,  one  gave  2-^  dol- 
lars and  the  other  3f  dollars,  what  did  they  give  for  the  whole 
barrel  ?  What  part  of  the  whole  value  did  each  pay  ?  What 
part  of  the  flour  should  each  have  ? 

70.  Two  men  hired  a  pasture  for  21  dollars.  One  kept 
his  horse  in  it  5|-  weeks,  and  the  other  7|  weeks ;  what 
ought  each  to  pay  1 

71.  What  part  of  7|- is  2|? 

72.  What  part  of  53|- is  13|7 

73.  What  part  of  107^  is  93^^  ? 

74.  What  part  of  3,840 J-^-  is  3^  1 

75.  What  part  of  f  is  -^  ? 
70.  Whatpartof  Uf  is  lf|  ? 

77.  What  part  of  28/9  is  i3f  ? 

78.  What  part  of  137^2_  ig  97_^3_  1 

79.  What  part  of  387/y  is  -j-f^  • 

XXIV.  1.*  If  I  of  a  gallon  of  brandy  co.t  $0.75,  wliat 
is  that  a  gallon  1 

*  See  First  Lessons,  Sect.  VI.  and  XI 


72  ARITHMETIC.  Part  1. 

%  If  I  of  a  ton  of  hay  cost  $13,375,  what  is  that  a  ton  \ 

3.  If  ^  of  a  yard  of  cloth  cost  $2,875  what  is  that  a 
yard? 

4.  If  i  of  a  hhd.  of  brandy  cost  $27.00,  what  will  1  hhd. 
cost  at  that  rate  1 

5.  A  merchant  bought  ^  of  a  pipe  of  brandy  for  $38.56  ; 
what  would  the  whole  pipe  come  to  at  that  rate  ? 

6.  A  smith  bought  |  of  a  ton  of  iron  for  $12.43;  what 
would  a  ton  cost  at  that  rate  ? 

7.  A  merchant  owned  yV  ^^  ^  ship's  cargo,  and  his  share 
was  valued  at  $8,467.00 ;  what  was  the  whole  ship  valued  at  ? 

8.  A  gentleman  owned  stock  in  a  bank  to  the  amount  of 
$8,642.00,  which  was  ^V  of  the  whole  stock  in  the  bank ; 
what  was  the  whole  stock  1 

9.  A  gentleman  lost  at  sea  $4,843.67,  which  was  ^^ 
of  his  whole  estate  ;  how  much  was  his  whole  property 
worth  ? 

10.  A  gentleman  bought  stock  in  a  bank  to  the  amount 
of  $873.14,  which  was  ^tt  of  the  value  of  his  whole  proper- 
ty.    What  was  the  value  of  his  whole  property  1 

11.  A  man  bought  ^  of  a  bushel  of  corn  for  4  of  a  dollar  ; 
what  would  be  the  price  of  a  bushel  at  that  rate  ? 

12.  A  man  bought  ^  of  a  bushel  of  rye  for  i  of  a  dollar ; 
what  would  a  bushel  cost  at  that  rate  ? 

13.  A  man  sold  |  of  a  yard  of  cloth  for  f  of  a  dollar ;  what 
would  a  yard  cost  at  that  rate  1 

14.  A  grocer  sold  |  of  a  gallon  of  wine  for  ■^\  of  a  dollar ; 
what  was  it  a  gallon  ? 

15.  A  grocer  sold  -3V  of  a  barrel  of  flour  for  -^K  of  a  dol- 
lar ;  what  was  it  a  barrel  ? 

16.  A  merchant  sold  )  of  a  ton  of  iron  for  19f  dollars  ; 
how  much  was  it  a  ton  T 

17.  A  merchant  sold  j\  of  a  hhd.  of  brandy  for  $11/7  ? 
how  much  was  it  per  hhd.  1 

18.  A  ship  of  war  having  taken  a  prize,  the  captain  re- 
ceived y\  of  the  prize  money.  His  share  aniounted  to 
$3^487^.     What  was  the  whole  prize  worth  ? 

19.  If  f  of  a  gallon  of  molasses  cost  20  cents,  what  will  I 
cost.  What  will  a  gallon  cost  ?  This  question  is  the  same 
as  the  following  :  If  2  quarts  of  molasses  cost  20  cents, 
whg.t  is  k  a  quart  1     How  much  a  gallon  ? 

20.  If  -}  of  a  gallon,  that  is  3  quarts,  of  molasses  cost  24 
cents,  what  w  ill  J-,  that  is  1  ([uart,  cost  1 


XXIV.  ARITHMETIC.  73 

21.  If  f  of  a  yard  of  cloth  cost  G  dollars,  what  cost  ^  1 
What  will  a  yard  cost  1 

22.  If  f  of  a  gallon,  that  is  3  pints,  of  wine  cost  90 
cents,  what  will  i,  that  is  1  pint,  cost  ?  What  will  a  gallon 
cost? 

23.  If  f  of  a  gallon  of  brandy  cost  95  cents,  what  will  | 
cost  1     What  will  a  gallon  cost  ? 

24.  If  I  of  a  yard  of  broadcloth  cost  $6.00,  what  will  j 
cost  1     What  will  a  yard  cost  7 

25.  If  f  of  a  box  of  lemons  cost  $2.40,  what  will  ^  cost  ? 
What  will  the  whole  box  cost  1 

26.  If  I  of  a  hhd.  of  molasses  cost  $16.00,  what  will  the 
whole  hogshead  cost  ? 

27.  A  man  travelljd  12  miles  in  yy  of  a  day  ;  how  far  did 
he  travel  in  j'^  of  a  day  1  How  far  would  he  travel  in  a  day 
at  that  rate  1 

28.  A  man  bought  f  of  a  barrel  of  flour  for  $4.85,  what 
would  be  the  price  of  a  barrel  at  that  rate  ? 

29.  A  man  being  asked  his  ag*3  answered,  that  he  was  24 
years  old  when  he  was  married,  and  that  he  had  lived  with 
his  wife  ^  of  his  whole  Vih.  Whr.t  part  of  Lis  whole  age  is 
24  years  I     What  was  his  age  ] 

30.  A  smith  bought  /g-  of  a  ton  of  Russia  iron  for  $25.35, 
what  would  be  the  price  of  a  ton  at  that  rate  1 

31.  Bought  I  of  a  yard  of  cloth  for  $5.00,  what  would  be 
the  price  of  a  yard  at  that  rate  ? 

32.  If  I  of  a  gallon  of  molasses,  that  is,  3  pints,  cost  17 
cents,  what  will  |,(1  pint,)cost  ?     What  will  a  gallon  cost  1 

33.  If  j^g  of  a  pound  of  snuff,  (5  ounces,)  cost  14  cents, 
what  cost  y'g  lb.,  (J  ounce.)  1 

34.  If  -^3  of  a  chaldron  of  coal  cost  $5,  what  cost  j\  1 
What  is  that  a  chaldron  ? 

35.  A  man  travelled  4  miles  in  |  of  an  hour ;  how  far 
would  he  travel  in  an  hour  at  that  rate  ? 

36.  Uj\  of  a  ship's  cargo  is  worth  $14,000,  what  is  the 
whole  cargo  worth  1 

37.  A  owns  if  of  a  coal  mine,  and  his  share  is  worth 
$3,500.     What  is  the  whole  mine  worth  1 

38.  If  j%\  of  the  stock  in  a  bank  is  worth  $63,275,  what 
is  the  whole  stock  worth  7 

39.  If  1|  yard  of  cloth  is  worth  $  11,  what  is  a  yard  worth  ? 

40.  If  2^  bushels  of  corn  is  worth  13  shillings,  what  is  a 
bushel  worth? 

7 


74  ARITHMETIC  Part  1. 

41  If  8/3  bushels  of  wheat  cost  $15,  what  is  it  a  bushel  1 
What  would  50  bushels  cost  at  that  rate  ? 

42.  A  man  sold  51/j  cwt.  of  sugar  for  $587  ;  what  would 
be  the  price  of  17?  cwt.  at  that  rate  1 

43.  If  f  of  1  lb.  of  butter  cost  f  of  a  dollar,  what  will  |  of 
1  lb.  cost "?     What  will  I  lb.  cost  ? 

44.  If  I  of  1  lb.  of  raisins  cost  fj  of  a  dollar,  what  will  J 
of  1  lb.  cost  ?     What  will  1  lb.  cost  ? 

45.  If  I  of  a  bushel  of  corn  cost  I  of  a  dollar,  what  is  that 
a  bushel  ? 

46.  If  jj  of  a  barrel  of  flour  will  serve  a  family  \^  of  a 
month,  how  long  will  one  barrel  serve  them  ?  How  long 
will  5  barrels  serve  them  1 

4,7.  If  4  of  a  yard  of  cloth  cost  4|  dollars,  what  is  that  a 
yard  1     What  will  17f  yards  cost  at  that  rate  ? 

48.  If  /g-  of  a  hhd.  of  wine  cost  30|  dollars,  what  will  be 
the  price  of  a  hhd.  at  that  rate  ? 

49.  If  3f  cwt.  of  iron  cost  814|-,  what  is  that  per  cwt.  ? 

50.  If  7f  lb.  of  butter  cost  $l-/i,  what  would  be  the  price 
of  27|  lb.  at  that  rate  ? 

51.  A  merchant  bought  a  piece  of  cloth  containing  24|- 
yards,  and  in  exchange  gave  32f  barrels  of  flour  ;  how  much 
flour  did  one  yard  of  the  cloth  come  to  1  How  much  cloth 
did  1  barrel  of  the  flour  come  to  ? 

52.  If  ^  of  a  yard  of  cloth  cost  f  of  a  pound,  what  will 
i\  of  an  ell  English  cost  ? 

53.  If  f  of  a  barrel  of  flour  cost  If  ^.,  what  will  43|  bar- 
rels cost  ? 

54.  A  person  having  f  of  a  vessel,  sells  f  of  his  share  for 
$8,400.00,  what  part  of  the  whole  vessel  did  he  sell  ?  What 
was  the  whole  vessel  worth  ? 

55.  If  I-  of  a  ship  be  worth  f  of  her  cargo,  the  cargo  being 
valued  at  2,000c£.,  what  is  the  whole  ship  and  cargo  worth  ? 

50.  If  by  travelling  12^  hours  in  a  day,  a  man  perform  a 
journey  in  7|-  days,  in  how  many  days  will  he  perform  it,  if 
he  travel  but  9^  hours  in  a  day  ? 

57.  If  5  men  mow  72f  acres  in  11|  days,  in  how  many 
days  will  8  men  do  the  same  ? 

58.  If  5  men  mow  72^  acres  in  11|-  days,  how  many 
jicres  will  they  mow  in  8|-  days  1 

59.  There  is  a  pole,  standing  so  that  f  of  it  is  in  the  water, 
I  as  much  in  the  mud  as  in  the  water,  and  7|  feet  of  it  is 
above  the  water.     What  is  the  whole  length  of  the  pole  ? 


XXIV.  ARITHMETIC.  75 

60.  A  person  having  spent  |  and  i  of  his  money  had 
$26|  left.     How  much  had  ho  at  first  1 

61.  Two  men,  A  and  B,  having  found  a  bag  of  money, 
disputed  who  should  have  it.  A  said  i,  |,  and  ^  of  the  money 
made  130  dollars,  and  if  B  could  tell  him  how  much  was 
in  it  he  should  have  it  all,  otherwise,  he  should  have  nothing. 
How  much  was  in  the  bag  1 

62.  45  is  f  of  what  number  1 

63.  486  is  yV  of  what  number  1 

64.  68  is  f  of  what  number  1 

65.  125  is  {f  of  what  number  1 

66.  376  is  If  of  what  number  ? 

67.  17  is  ^  of  what  number  ? 

68.  3  is  /o^  of  what  number  ? 

69.  68  is"  gVy  of  what  number  ? 

70.  253  is  yi!  5  of  what  number  ? 

71.  37  is  ifl  of  what  number  ? 

72.  6845  is  -j-\Vt  ^^  ^^^^  number  7 

73.  384  is  ViVVe  of  what  number  1 

74.  I  is  -f  of  what  number  1 

75.  2  is  3  Qf  what  number  ? 

76.  -f  is  I  of  what  number  1 

77.  j3_  ig  4  Qf  what  number  ? 

78.  1 1  is  ^  of  what  number  1 

79.  II  is  y\  of  what  number  1 

80.  1^  is  ^1  of  what  number  ? 

81.  -flf  is  y\  of  what  number  ? 

82.  If  is  III  of  what  number? 


83.  ^  is  ^V/t  of  what  number  ? 

84.  3f  is  ll  of  what  number  ? 

85.  14^  is  -^\  of  what  number  7 

86.  28|  is  -Vs  of  what  number  1 

87.  135||  is  -j\  of  what  number  1 

88.  384y\-  is  Yt  of  what  number  1 

89.  13fJ  is  fl^  of  what  number  1 

90.  Divide  13|f  by  f||-. 

91.  18|-4  is  II  of  what  number? 

92.  Divide  ISff  by  |^. 

93.  427|  is  y  of  what  number  ? 

94.  Divide  42|  by  2|,  that  is  y. 

95.  3S4y\  is  V  of  what  number  ? 

96.  Divide  384y\  by  3|  or  V- 

97.  42  is  I  of  what  number  ? 


76  ARITHMETIC.  Part  1. 

98.  How  many  times  is  f  contained  in  42  ? 

99.  Divide  42  by  f . 

100.  3j%  is  f  of  what  number  1 

101.  How  many  times  is  f  contained  in  3-^^  1 

102.  Divide  3fV  by  f 

103.  13|  is  y  of  what  number  ? 

104.  How  many  times  is  2f  or  y  contained  in  13|- 1 

105.  Divide  13|  by  2f . 

106.  A  merchant  sold  a  quantity  of  goods  for  $252.00, 
which  was  f  of  what  it  cost  him  1  How  much  did  it  cosK 
him,  and  how  much  did  he  gain  1 

107.  A  mercliant  sold  a  quantity  of  goods  for  $243.00, 
by  which  he  gained  |  of  the  first  cost.  What  was  the  first 
cost,  and  how  much  did  he  gain  7 

Note.  If  he  gained  \  of  the  first  cost,  $243.00  must  be 
I  of  the  first  cost. 

108.  A  merchant  sold  a  quantity  of  goods  for  $3,846.00, 
by  which  bargain  he  gained  ^  of  the  first  cost.  What  was 
the  first  cost,  and  how  much  did  he  gain  7 

109.  A  merchant  sold  a  hhd.  of  wine  for  $108.43,  by 
which  bargain  he  gained  \  of  the  first  cost.  What  was  the 
first  cost  per  gallon  1 

110.  A  merchant  sold  a  bale  of  cloth  for  $347.00,  by 
which  he  gained  -^\  of  what  it  cost  him  1  How  much  did  it 
cost  him,  and  how  much  did  he  gain  1 

Note.  If  he  gained  -^\  of  the  first  cost,  $347.00  must  be 

i|  of  the  first  cost. 

111.  A  merchant  sold  a  quantity  of  flour  for  $147.00,  by 
which  he  gained  |  of  the  cost.  How  much  did  it  cost,  and 
how  much  did  he  gain  1 

112.  A  merchant  sold  a  quantity  of  goods  for  $6,487.00, 
by  which  he  gained  -^-j  of  the  cost.    How  much  did  he  gain? 

113.  A  merchant  sold  a  quantity  of  goods  for  $187.00  by 
which  he  lost  i  of  the  first  cost.  How  much  did  it  cost,  and 
how  much  did  he  lose '? 

Note.  If  he  lost  \  of  the  cost,  $187.00  must  be  f  of  the 
cost. 

114.  A  merchant  sold  a  quantity  of  molasses  for  $258.00, 
by  which  he  lost  |  of  the  cost.     How  much  did  it  cost,  and 

how  much  did  he  lose  1 


XXIV.  ARITHMETIC.  77 

115.  A  merchant  sold  a  quantity  of  goods  for  $948.00,  by 
which  he  lost  -^j  of  the  cost.     How  much  did  he  lose  1 

116.  A  merchant  sold  3  hhds.  of  molasses  for  $07.23,  by 
which  he  lost  -^^  of  the  first  cost.  How  much  did  he  lose  1 
How  much  on  a  gallon  1 

117.  A  merchant  sold  93  yards  of  cloth  for  $527.43,  by 
which  he  lost  -^j  of  the  cost.  How  much  did  he  lose  on  a 
yard  ? 

118.  A  merchant  sold  a  quantity  of  goods  so  as  to  gain 
$43,  which  was  f-  of  what  the  goods  cost  him.  How  much 
did  they  cost  ? 

119.  A  merchant  sold  a  quantity  of  goods  for  $273.00,  by 
which  he  gained  10  per  cent,  on  the  first  cost.  How  much 
did  they  cost  ? 

Note.  10  per  cent,  is  10  dollars  on  a  100  dollars,  that  is,  j^^. 
10  per  cent,  of  the  first  cost  therefore  is  yVV  of  the  first  cost. 
Consequently  $273.00  must  be  \^  of  the  first  cost. 

120.  A  merchant  sold  a  quantity  of  goods  for  $135.00, 
by  which  he  gained  13  per  cent.  How  much  did  the  goods 
cost,  and  how  much  did  he  gain  ? 

121.  A  merchant  sold  a  quantity  of  goods  for  $3,875  by 
which  he  gained  65  per  cent.  How  many  dollars  did  he 
gain  1 

122.  A  merchant  sold  a  quantity  of  goods  for  $983.00,  by 
which  he  lost  12  per  cent.  How  much  did  the  goods  cost, 
and  how  much  did  he  lose  ? 

Note.  If  he  lost  12  per  cent.,  that  is  ^,  he  must  have 
sold  it  for  -/^  of  what  it  cost  him. 

123.  A  merchant  sold  3  hhds.  of  brandy  for  $248.37,  by 
which  he  lost  25  per  cent.  How  much  did  the  brandy  cost 
him,  and  how  much  did  he  lose  ? 

124.  A  merchant  sold  a  quantity  of  goods  for  $87.00  more 
than  he  gave  for  them,  by  which  he  gained  13  per  cent,  of 
the  first  cost.  What  did  the  goods  cost  him,  and  how  much 
did  he  sell  them  for  1 

Note.  Since  13  per  cent,  is  V^,  $87  must  be  -J^  of  the 
first  cost. 

125.  A  merchant  sold  a  quantity  of  goods  for  $43.00  more 
than  they  cost,  and  by  doing  so  gained  20  per  cent.  How 
much  did  the  goods  cost  him  1 

7* 


78  ARITHMEl'IC.  Part  1 

126.  A  merchant  sold  a  quantity  of  goods  for  $137.00  less 
than  they  cost  him,  and  by  doing  so  lost  23  per  cent.  How 
much  did  the  goods  cost,  and  how  much  did  he  sell  them 
for? 

127.  A  has  tea  which  he  sells  B  for  lOd.  per  lb.  more  than 
it  cost  him,  and  in  return  B  sells  A  cambrick,  which  cost 
him  lOs.  per  yd.,  for  12s.  6d.  per  yard.  The  gain  on  each 
was  in  the  same  proportion.  What  did  A's  tea  co^t  him 
per  lb.  7 

Note.  B  gains  2s.  6d.  on  a  yard,  which  is  \  of  the  first 
cost,  consequently  lOd.  must  be  \  of  the  first  cor;t  of  the 
tea? 

128.  C  has  brandy  which  he  sells  to  D  for  20  cents  per 
gal.  more  than  it  cost  him  ;  and  D  sells  C  molasses  which 
cost  23  cents  per  gal.  for  32  cents  per  gal.,  by  which  D  gams 
in  the  same  proportion  as  C.  How  much  did  C's  brandy 
cost  him  per  gal.  ? 

129.  A  man  being  asked  his  age,  answered,  that  if  to  his 
age  ^  and  -]-  of  his  age  be  added,  the  sum  would  be  121. 
What  was  his  age  ? 

130.  A  man  having  put  a  sum  of  money  at  interest  at  6 
per  cent.,  at  the  end  of  1  year  rect^ived  ]3  dollars  for  interest. 
What  was  the  principal  ? 

Note.  Since  6  per  cent,  is  y^  of  the  whole,  13  dollars 
must  be  y|  ^  of  the  principal. 

131.  What  sum  of  money  put  at  interest  for  1  year  will 
gain  57  dollars,  at  6  per  cent.  ? 

1.32.  A  man  put  a  sum  of  money  at  interest  for  1  year,  at 
6  per  cent.,  and  at  the  end  of  the  year  he  received  for  prin- 
cipal and  interest  237  .dollars.     What  was  the  principal  ? 

Note.  Since  6  per  cent,  is  y^^^,  if  this  be  added  to  the 
principal  it  will  make  |f|,  therefore  $237  must  be  |f|  of 
the  principal.  When  the  interest  is  added  to  the  principal 
the  whoie  is  called  the  amount. 

133.  What  sum  of  money  put  at  interest  at  6  per  cent. 
will  gain  $53  in  2  years  ? 

Note.  6  per  cent,  for  1  year  will  be  12  per  cent,  for  2 
years,  3  per  cent,  for  6  months,  1  per  cent,  for  2  months, 
&c. 

134.  What  sum  of  money  put  at  interest  at  6  per  cent 
will  gain  $97  in  one  year  and  6  months  ? 


XXIV.  ARITHMETIC.  79 

135.  What  sum  of  money  put  at  interest  at  6  per  cent, 
will  amount  to  $394  in  1  year  and  8  months  1 

136.  What  sum  of  money  put  at  interest  at  7  per  cent, 
will  amount  to  =£183  in  1  year  1 

137.  What  sum  of  money  put  at  mterest  at  8  per  cent, 
will  amount  to  $137  in  2  years  and  6  months  1 

138.  Suppose  I  owe  a  man  $287  to  be  paid  in  one  year 
without  interest,  and  I  wish  to  pay  it  now ;  how  much  ought 
I  to  pay  him,  when  the  usual  rate  is  6  per  cent.  1 

Note.  It  is  evident  that  I  ought  to  pay  him  such  a  sum, 
as  put  at  interest  for  1  year  will  amount  to  $287.  The 
question  therefore  is  like  those  above.  This  is  sometimes 
called  discount. 

139.  A  man  owes  $847  to  be  paid  in  6  months  without 
interest,  what  ought  he  to  pay  if  he  pays  the  debt  now,  al 
lowing  money  to  be  worth  G  per  cent,  a  year  ? 

140.  A  merchant  being  in  want  of  money  sells  a  note  of 
$100,  payable  in  8  months  without  interest.  How  much 
ready  money  ought  he  to  receive,  when  the  yearly  interest 
of  money  is  6  per  cent.  1 

141.  According  to  the  above  principle,  what  is  the  differ 
ence  between  the  interest  of  $100  for  1  year,  at  6  per  cent, 
and  the  discount  of  it  for  the  same  time  ? 

142.  What  is  the  difference  between  the  interest  of  $500 
for  4  years  at  6  per  cent.,  and  the  discount  of  the  same  sum 
for  the  same  time  1 


Miscellaneous  Examples. 

In  measuring  surfaces,  such  as  land,  &,c.  a  square  is  used 
as  the  measure  or  unit.  A  square  is  a  figure  with  four  equal 
sides,  and  the  four  corners  or  angles  equal.  The  square  is 
used    because  it  is  more  convenient     A  B 

ior  a  measure  than  a  figure  of  any 
other  form.  The  figure  a  b  c  d  is  a 
square.  The  sides  are  each  one  inch, 
consequently  it  is  called  a  square 
inch.  A  figure  one  foot  long  and  one 
foot  wide  is  called  a  square  foot ;  a 
figure  one  yard  long  and  one  yard 
wide  is  called  a  square  yard,  &c. 


so  ARITHMETIC.  Fart  J. 

1.  If  a  figure  one  inch  long  and  one  inch  wide  contains 
one  square  inch,  how  many  square  inches  does  a  figure  one 
inch  wide  and  two  inches  long  contain  ?  How  many  square 
inches  does  a  figure  one  inch  wide  and  three  inches  long 
contain  1  Four  inches  long  1  Five  inches  long  ?  Seven 
inches  long  ? 

2.  In  a  figure  8  inches  long  and  1  inch  wide,  how  many 
square  inches  ?  How  many  square  inches  does  a  figure  8 
inches  long  and  2  inches  wide  contain  1  3  inches  wide  1  4 
inches  wide  ?    5  inches  wide  ?    8  inches  wide  1 

3.  If  a  figure  1  foot  wide  and  1  foot  long  contains  1  square 
foot,  how  many  square  feet  does  a  figure  I  foot  wide  and  2 
feet  long  contain  ?  How  many  square  feet  does  a  figure  1 
foot  wide  and  3  feet  long  contain  1  5  feet  long  ?  9  feet 
long  ?    15  feet  long  1 

4.  In  a  figure  9  feet  long  and  1  foot  wide,  how  many 
square  feet  1  How  many  square  feet  does  a  figure  9  feet 
long  and  2  feet  wide  contain  ?    3  feet  wide  1    5  feet  wide  1 

7  feet  wide  ?    9  feet  wide  1 

5.  How  many  square  inches  does  a  figure  13  inches  long 
and  I  inch  wide  contain  1    2  inches  wide  1    3  inches  wide  1 

8  inches  wide  1 

6.  How  many  square  feet  does  a  figure  16  feet  long  and  1 
foot  wide  contain  ?  2  feet  wide  ?  3  feet  wide  %  5  feet 
wide  1    8  feet  wide  1    13  feet  wide  ? 

In  the  above  examples  supply  yards,  rods,  furlongs,  and 
miles,  instead  of  inches  and  feet,  and  perform  them  again. 

7.  What  rule  can  you  make  for  finding  the  number  of 
square  inches,  feet,  yards,  &c.  in  any  rectangular  figure  ? 

Note.  A  figure  with  four  sides,  which  has  all  its  angles 
alike  or  right  angles,  is  called  a  rectangle,  and  a  rectangle  is 
called  a  square  when  all  the  sides  are  equal. 

8.  How  many  square  feet  in  a  room  18  feet  long  and  13 
feet  wide  1 

9.  How  many  square  feet  in  a  piece  of  land  143  feet  long 
and  97  feet  wide  ? 

10.  How  many  square  rods  in  a  piece  of  land  28  rods 
long  and  7  rods  wide  1 

11.  A  piece  of  land  that  is  20  rods  long  and  8  rods  wide^ 
or  in  any  other  form  containing  the  same  surface,  is  called 
an  acre.     How  many  square  rods  in  an  acre  1 


XXIV.  ARITHMETIC.  81 

12.  How  wide  must  a  piece  of  land  be  that  is  17  roda 
long  to  make  an  acre  ? 

13.  How  many  square  inches  in  a  square  foot ;  that  is,  in 
a  figure  that  is  1'2  inches  long  and  12  wide  ? 

14.  How  much  in  length,  that  is  8  inches  wide,  will  make 
a  square  foot '? 

15.  How  many  square  feet  in  a  square  yard  ? 
IG.  How  many  square  yards  in  a  square  rod  1 

17.  How  many  square  inches  in  a  square  yard  ? 

18.  A  piece  of  land  20  rods  long  and  2  rods  wide,  or  in 
any  other  form  which  contains  the  same  surface,  is  called  a 
rood.     How  many  square  rods  in  a  rood  1 

19.  How  many  roods  make  an  acre  ? 

20.  Find  the  numbers  for  the  following  table. 


SQUARE    MEASURE. 

square  inches  make       1  square  foot 

square  feet  1  square  yard 

square  yards  or  >  1  square  rod, 

square  feet  )  perch,  or  pole 

square  rods  1  rood 

roods  1  acre 

21.  How  many  square  inches  in  a  square  rod  1 

22.  How  many  square  yards  in  an  acre  1 

23.  How  many  square  inches  in  an  acre  1 

24.  How  many  square  feet  in  1728  square  inches  1 

25.  In  2SG  square  poles  how  many  acres  ? 

26.  In  201,283,876  square  inches,  how  many  acres? 

27.  How  many  square  rods  in  a  square  mile  ?  ^, 

28.  How  many  acres  in  a  square  miles  ? 

29.  The  whole  surface  of  the  globe  is  estimated  at  about 
(98,000,000  square  miles.  How  many  acres  on  the  surface 
of  the  globe  1 

30.  How  many  square  inches  in  a  board  15  inches  wide 
and  1 1  feet  long  ?     How  many  square  feet  ? 

31.  How  many  acres  in  a  piece  of  land  183  rods  long 
and  97  rods  wide  ? 

32.  How  many  square  inches  in  a  yard  of  carpeting  that  is 
2  ft.  3  in.  wide  1  How  many  yards  of  such  carpeting  will  it 
take  to  cover  a  floor  19  ft.  4  in.  long  and  17  ft.  2  in.  wide  I 


82  ARITHMETIC.  Part  1. 

To  measure  solid  bodies,  such  as  timber,  wood,  &c.,  it  is 
necessary  to  use  a  measure  that  has  three  dimensions, 
length,  breadth,  and  depth,  height,  or  thickness.  For  this  a 
measure  is  used  in  which  all  these  dimensions  are  alike. 
Take  a  block,  for  example,  and  make  it  an  inch  long,  an 
inch  wide,  and  an  inch  thick,  and  all  its  corners  or  angles 
alike  ;  this  is  called  a  solid  or  cubic  inch  ;  so  a  block  made 
in  the  same  way  having  each  of  its  dimensions  one  foot,  is 
called  a  solid  or  cubic  foot. 

33.  If  a  block  1  inch  wide  and  1  inch  thick  and  1  inch 
long  contains  1  solid  inch,  how  many  solid  inches  does  such 
a  block  that  is  2  inches  long  contain  1  3  inches  long  1  4 
inches  long  1  5  inches  long  1   8  inches  long  ? 

34.  How  many  solid  inches  does  a  block  that  is  1  foot 
long,  1  inch  thick,  and  1  inch  wide  contain  ?  How  many 
inches  does  such  a  block  that  is  2  inches  wide  contain  ?  3 
inches  wide  1  4  inches  wide  ?  5  inches  wide  ?  8  inches 
wide  t 

35.  How  many  solid  inches  does  a  block  2  inches  long,  2 
inches  wide,  and  1  inch  thick  contain  1   2  inches  thick  ? 

36.  How  many  solid  inches  does  a  block  4  inches  long,  3 
inches  wide,  and  1  inch  thick  contain  1  2  inches  thick  1  3 
inches  thick  ? 

37.  How  many  cubic  inches  in  a  block  10  inches  long,  8 
inches  wide,  and  1  inch  thick  ?  2  inches  thick  ?  3  inches 
thick  ?  5  inches  thick  ?  7  inches  thick  ? 

38.  How  many  cubic  inches  in  a  block  18  inches  long, 
13  inches  wide,  and  1  inch  thick  ?  5  inches  thick  ?  11 
inches  thick  ? 

In  the  above  examples  supply  feet  instead  of  inches, 
and  do  them  over  again. 

39.  What  rule  can  you  make  for  finding  the  number  of 
solid  inches  or  feet  in  any  regular  solid  body  ? 

40.  How  many  solid  inches  in  a  block  12  inches  long,  12 
inches  wide,  and  12  inches  thick  ;  that  is,  in  a  solid  foot  ? 

41.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet 
high,  or  in  any  other  form  containing  an  equal  quantity,  is 
called  a  co7'd  of  wood.     How  many  solid  feet  in  a  cord  ? 

4'^.  Find  the  numbers  for  the  following  table. 


XXV.  ARITHMETIC.  83 

SOLID    OR    CUBIC    MEASUREr 

solid  inches  make  1  solid  foot 

solid  feet  1  cord  of  wood 

40  solid  feet  of  round  timber,  or  )  -.  .  ,     . 

50  solid  feet  of  hewn  timber         /  ^  ^^"  ""'  ^^^^ 

43.  How  many  solid  inches  in  a  cord  ?  ' 

44.  How  many  solid  inches  in  a  ton  of  hewn  timber  ? 

45.  In  468,374  solid  inches,  how  many  solid  feet  ? 

46.  How  many  feet  of  timber  in  a  stick  28  feet  long  and 
11  inches  square  ? 

47.  How  many  tons  of  timber  in  2  sticks,  each  25  feet 
long,  15  inches  wide,  and  11  inches  thick  ? 

48.  A  pile  of  wood  4  feet  square  and  1  foot  long,  or  a  pile 
containing  16  solid  feet  is  called  1  foot  of  wood.  How 
many  such  feet  in  a  cord  ? 

49.  How  many  solid  feet  of  wood  in  a  pile  5  feet  wide,  3 
feet  high,  and  23  feet  long  ?  How  many  feet  of  wood  ? 
How  many  cords  ? 

A  few  more  examples  of  this  kind  will  be  found  in  deci- 
mals. 


DECIMAL  FRACTIONS. 

XXV.     In  the  following  numbers,  write  the  fractional 
part  in  the  form  of  decimals. 

1.  Twenty-seven  and  six  tenths,  27^q.  Ans.  27.6. 

2.  Fourteen  and  seven  hundredths.    14^^^. 

Ans.  14.07. 

3.  One  hundred  twenty-three,    and  eight   thousandths. 
123.n^.  A?is.  123.008. 

4.  One  hundred  and  eight,  and  five  tenths.  108y^^. 

5.  Seventy-three,  and  nine  hundredths.  73yJ^ 

6.  Four,  and  six  thousandths,  "^j-^-qj^- 

7.  Sixteen,  and  one  thousandth.  16y^^. 

8.  Six  tenths.  j\. 

9.  Five  hundredths,  -t^t^- 


10.  Seven  thousandths,  j/^tq. 

11.  Two  ten  thousandths,  jo^^^* 


84 


ARITHMETIC. 


Parti. 


12.  Three,  and  four  tenths  and  two  hundredths.  3^*^  and 


loo* 

13. 
14. 
15. 

16. 
17. 

18. 
19. 


y*g-  are  how  many  hundredths  1 
j\  andp^-Q-  are  how  many  hundredths  1 
j\  are  how  many  thousandths  ] 
-yIq  are  how  many  thousandths  1 


To  and 


roo- 

3  85 
000 


and 


are  how  many  thousandths  1 


1000 

Write  Tf'^g-  in  the  form  of  a  decimal. 

■f^  are  how  many  ten-thousandths  1 
30.  yf  0  are  how  many  ten-thousandths  ? 
21.  y/o^  are  how  many  ten-thousandths  ? 


QO        2 


0  o> 


and  j^l^-^ 


ten-thou- 


are  how  many 

sandths  ? 

23.  Write  ^WVo  in  the  form  of  a  decimal  1 

Write  the  fractions  in  the  following  numbers  in  the  form 

of  decimals. 


24. 
25. 
26. 

27. 
28. 
29. 


in  2  3 

21-1-8-2-. 
■^-^  10  0  0* 

1  0_5JL36_ 
■^'*1  0  0  0  0* 

IJl 38746 

^^'^rooooo" 

l-43_ 

*  1  0  0  0* 
17  57  3 

^'Tooo-^* 


30. 
31. 
32. 
33. 
34. 
35. 


95, 


IQQ 4_7 

87— i-OA_. 

^'  1  0  0000 

10  0  0* 
QQ   6  0  04 

•^^TooooTTo"* 

J5_o_5_P_L. 
1  0  n  0  00* 

807 
10000' 


Change  the  decimals  in  the  following  numbers  to  com- 
mon fractions  and  reduce  them  to  their  lowest  terms. 


36. 
37. 
38. 
39. 
40. 
41. 
42. 
43. 
44. 


42..5. 

84.25. 

9.8. 

137.16. 

25.125. 

18.625. 

11.8642. 

103.90064. 

72.0065. 


45. 
46. 
47. 
48. 
49. 
50. 
51. 
52. 
53. 


4.00025. 

13.0060058. 

0.75. 

0.3125. 

.075. 

.00123. 

.00015. 

.000106. 

.1500685. 


XXVI.  1.  A  man  purchased  a  barrel  of  flour  for  $7.43. ; 
5  gallons  of  molasses  for  $1,625;  3  gallons  of  wine  for 
$4.87  ;  4  gallons  of  brandy  for  $7  ;  7  lbs.  of  sugar  for 
$0.95 ;  and  3  gallons  of  vinegar  for  $0.42.  What  did  the 
whole  amount  to  ? 

2.  How  many  bushels  of  corn  in  4  bags,  the  first  contain- 
ing ^tV  ^"^^^els  ;  the  second,  3^^;  the  third,  3-^if^ ;  and 
thefourth,  4f,^? 

Note.     Write  the  fractions  in  the  form  of  decimals. 


XXVII.  DECIMALS.  85 

3.  A  man  bought  four  loads  of  hay,  the  first  containing 
17f  cvvt. ;  the  second,  19^  cwt.  ;  the  thirl,  24  f  cwt. ;  and 
the  fourth,  14^  cvvt.      How  many  cwt.  in  the  whole  1 

Note.  In  all  the  examples  under  the  head  of  decimals, 
change  the  fractions  and  parts  to  decimals. 

4.  A  man  raised  wheat  in  five  fields,  in  the  first,  47-^'^ 
bushels  ;  in  the  second,  Oi^^  "?  in  the  third,  S7i|- ;  in  the 
fourth,  14a|i  ;  and  in  the  fifth  387  bushels,  ilow  many 
bushels  in  the  whole  ? 

5.  A  man  bought  a  load  of  hay  for  6'^j£. ;  a  load  of  oats 
^^^  "^jo^- '  ^  bushels  of  corn  for  ^l£.  ;  and  a  load  of  wood 
for  2-.\£.     How  much  did  the  whole  come  to? 

6.  Add  together  the  following  numbers,  38-i| ;  1386^^^ ; 
7006;^/,V;  /A6_  ;  8;  and  460||. 

7.  From  a  piece  of  cloth  containing  47|  yards,  a  mer- 
chant sold  23/j.     How  much  remained  unsold  1 

8.  A  man  owing  $253  paid  §187.375,  how  much  did  he 
then  owe  1 

9.  A  man  owing  342,%4_£.  paid  187 /y^.  How  much 
did  he  then  owe  ? 

10.  A  merchant  sold  a  barrel  of  flour  for  2/3^. ;  5  gal- 
lons of  molasses  for  \^£, ;  and  6  gallons  of  wine  for  2i-|<£. 
In  pay  he  received  a  load  of  wood  worth  2j\£.  and  2  bush- 
els of  wheat,  worth  ^£.  and  the  rest  in  money  ;  how  much 
money  did  he  receive  ? 

11.  From  183fc£.  take  87f^. 

12.  From  $382  take  $48.25. 

13.  From  ll53f  lb.  take  684-i-Vb. 

14.  From  373-  tor.s  'ake  28 j\  tons. 


Multiplication  of  Decimals. 

XXVIT.     1.  A  man  bought  5  barrels  of  pork,  at  $17.43 
per  barrel ;  how  much  did  it  come  to  1 

2.  What  cost  8  yards  of  cloth,  at  $7,875  per  yard  ? 

3.  How  many  bushels  of  meal  in  14  sacks,  containing 
4.37  bushels  each  1 

4.  How  much  hay  in  8  loads,  containing  24.35  cwt.  each  ? 

5.  How  much  cotton  in  17  bales,  containing  4|  cwt.  each  ? 

6.  How  many  cwt.  of  hay  in  14  loads,  containing  23.25 
cwt.  each  ? 

8 


86  ARITHMETIC.  Part  1. 

7.  Multiply  42.62  by  3S. 

8.  Multiply  137.583  by  17. 

9.  Multiply  13.946  by  58. 

10.  Multiply  2.5837  by  15. 

11.  Multiply  .464  by  27. 

12.  Multiply  .0038  by  9. 

13.  If  a  barrel  of  flour  cost  $5,  what  cost  .6  of  a  barrel  1 

14.  At  $90  per  hhd.,  what  cost  .7  hhd.,  that  is,  ^^  of  a 
hdd.  ] 

15.  At  845  per  hhd.,  what  cost  .8  hhd.,  that  is,  -^  of  a 
hhd.  of  gin  ? 

16.  At  $20  per  hhd.,  what  cost  2.9  hhds.,  that  is  2^  hhds. 
of  molasses  1 

17.  At  825  per  ton,  what  cost  7.6  tons  of  hay  1 

18.  At  $95  per  ton,  what  cost  3.7  tons  of  iron  1 

19.  At  832  per  ton,  what  cost  14.25  tons  of  logwood  ? 

20.  At  $220  per  ton,  what  cost  19.47  tons  of  hemp  ? 

21.  At  857  per  ton,  what  cost  3.5  tons  of  alum  ? 

22.  At  845  per  thousand,  what  cost  2.5  thousands  of 
staves  ? 

23.  What  is  .5  of  128? 

24.  What  is  .25  of  856? 

25.  What  is  .125  of  856? 

26.  What  is  .287  of  2487? 

27.  Multiply  2487  by  .287. 

28.  Multiply  4306  by  3.5. 

29.  Multiply  87  by  2.8. 

30.  Multiply  1864  by  3.25. 

31.  Multiply  30067  by  1.3873. 

32.  Multiply  10372  by  6izi=6.5. 

33.  Multiply  468  by  7-iz=7.25. 

34.  Multiply  46800  by  13|. 

35.  Multiply  36038  by  1^. 

36.  Multiply  130407  by  5^^^. 

37.  At  .3  of  a  dollar  a  gallon,  what  cost  .2  of  a  gallon  ol 
molasses  ? 

38.  What  is  .2  of  .3,  that  is  ,?^  of -j-^  ? 

39.  Multiply  .3  by  .2. 

40.  At  8-90  per  gallon,  what  cost  .4  of  a  gal.  of  wine  ? 

41.  At  8.25  per  lb.  what  cost  2.8  lb.  of  butter  ? 

42.  At  8.36  per  lb.,  what  cost  4.5  lb.  of  sperm  candles  ? 

43.  At  $.47  per  piece,  what  cost  4.3  pieces  of  nankin  ? 

44.  At  85.37  per  yard,  what  cost  7.4  yards  of  cloth  ? 


XXVII.  DECIMALS,  87 

45.  At  $13.50  per  bbl.,  what  cost  14f  bbls.  of  pork? 

46.  At  $25.45  per  ton,  what  cost  18|  tons  of  liay  ? 

47.  At  $140..50  per  ton,  what  cost  13|  tons  of  potashes  ? 

48.  If  an  orange  is  worth  $.06,  what  is  .3  of  an  orange 
worth  ? 

49.  If  a  bale  of  cotton  contains  4.37  cwt.,  what  is  .45  of  a 
bale? 

50.  Multiply  4.5  by  2.3. 

51.  Multiply  13.43  by  1.4. 

52.  Multiply  43.25  by  .8. 

53.  Multiply  284.43  by  1.02. 

54.  Multiply  18.325  by  1.38. 

55.  Multiply  6.4864  by  2.03. 

56.  Multiply  14.00643  by  .5. 

57.  Multiply  3.400702  by  1.003. 

58.  Multiply  1.006  by  .002. 

59.  Multiply  1.0007  by  .0003. 

60.  Multiply  .3  by  .2. 

61.  Multiply  .04  by  .2. 

62.  Multiply  .003  by  .01. 

63.  Multiply  .0004  by  .025. 

64.  Mi'ltiply  .0107  by  .00103. 

65.  Multiply  1.340068  by  1.003084. 


Miscellaneous  Examples. 

1.  At  $12  per  cwt.  what  cost  5  cwt.  3  qrs.  of  sugar  ? 
Note.     5  cwt.  3  qrs.  is  5f  cwt.,  that  is  5.75  cwt. 

2.  At  $25  per  cwt.,  what  cost  37  cwt.  3  qrs.  14  lb.  of  to- 
Dacco  ? 

Note.  The  quarters  and  pounds  may  first  be  reduced  to 
a  common  fraction  and  then  to  decimals.  3  qrs.  14  lb.  are 
98  lb.,  that  is  j\\  of  1  cwt.,  and  -rV5z:r.875  ;  therefore,  37 
cwt.  3  qrs.  14  lb.  "is  equal  to  37.875  cwt. ;  this  multiplied  by 
25  gives  $946,875. 

3.  What  cost  5  cwt.  2  qrs.  19  lb.  of  raisins,  at  $11  per 
cwt.  ? 

4.  What  cost  13  cwt.  1  qr.  15  lb.  of  iron,  at  $4.27  per 
cwt.  1 


88  ARITHMETIC  Part  1, 

Note.     13  cvvt.  1  qr.  15  Ib-z^ia^-Va   cwt.z=:13.383+cwt- 

This  multiplied  by  $4.27  gives  $57.14541.  Observe,  that 
there  must  be  as  many  decimal  places  in  the  product  as  in 
the  multiplicand  and  multiplier  together.  In  this  instance 
there  are  five  places.  It  is  not  necessary  to  notice  any  thing 
smaller  than  mills  in  the  result,  therefore  857.145  will  be 
sufficiently  exact  for  the  answer. 

5.  What  cost  12  cwt.  0  qrs.  19  lb.  of  rice,  at  $3.28  per 
cwt.  1 

6.  What  cost  13  cwt.  2  qrs.  4  lb.  of  hops,  at  $5.75  per 
cwt. 

7.  W  hat  cost  3  hhds.  43  gal.  of  wine,  at  $98  per  hhd.  1 

Note.  3  hhds.  43  gal.  is  3  Jf  hhds. ;  this  reduced  to  a  de- 
cimal is  3.683  hhds.,  nearly. 

8.  What  cost  17  hhds.  18  gal.  of  molasses,  at  $23.25  per 
hhd.  1 

9.  What  cost  13  hhds.  53  gal.  of  gin,  at  $47,375  per 
hhd.? 

10.  What  cost  4  hhds.  27  gal.  3  qts.  of  brandy,  at  $108.42 
per  hhd.  ? 

11.  Express  in  decimals  of  an  cwt.  the  quarters,  pounds, 
and  ounces  in  the  following  numbers : — 3  cwt.  2  qrs.  22  lb. ; 
17  cwt.  1  qr.  11  lb.  5  oz. ;  4  cwt.  0  qr.  16  lb.  3  oz. 

12.  Express  in  decimals  of  a  hogshead  the  gallons,  quarts, 
pints,  &,c.  in  the  following  numbers  : — 43  hhds.  17  gal.  2 
qts  ;  14  gal.  6  qts.  1  pt.  ;  7  hhds.  0  gal.  3  qts.  1  pt. 

13.  What  cost  8  gal,  3  qts.  1  pt.  of  gin,  at  $0.43  pef  gal.  1 

14.  What  cost  17^  lb.  13  oz.  of  sugar,  at  $0.12  per  lb.  ? 

15.  What  cost  231b.  7  oz.  of  sugar,  at  $11.43  per  cwt.  ? 

16.  What  cost  11  gals.  2  qts.  of  brandy,  at  the  rate  of 
$98.48  per  hhd.  ? 

17.  What  cost.  17  yds.  3  qrs.  2  nls.  of  broadcloth,  at  $7.25 
per  yard  ? 

18.  What  cost  2  qrs.  3  nls.  of  broadcloth,  at  $6.42  pei 
yard  1 

Express  the  fractions  in  the  following  examples  in  deci- 
mals. 

19.  What  part  of  1  yd.  is  3  qrs.  2  nls.  ? 

20.  What  part  of  1  yard  is  1  qr.  3  nls.  ? 

21.  What  part  of  1  lb.  Avoirdupois  is  13  oz.? 

22.  What  part  of  1  qr.  is  17  lb.  1 

23.  What  part  of  1  qr.  is  13  lb.  5  oz.  1 


XXVII.  DECIMALS.  S9 

24.  What  part  of  a  day  is  6  hoars  '? 

25.  What  part  of  a  day  is  16  h.  25  min.  1 

20.  What  part  of  a  day  is  13  h.  42  min.  11  sect 

27.  What  part  of  an  hour  is  47  min.  ? 

28.  What  part  of  an  hour  is  38  min.  47  sec.  1 

29.  What  part  of  a  rod  is  13  ft.  ? 

30.  What  part  of  I  ft.  is  2  in.  ? 
31  What  part  of  I  ft.  is  7  in.  ? 

32.  Wliat  part  of  a  rod  is  7  ft.  4  in.  ? 

33.  What  part  of  a  mile  is  7  rods,  13  ft.  t 

34.  What  part  of  l£.  is  13s.  6d.  ? 

35.  What  part  of  Is.  is  5d.  1  qr? 

3(5.  What  part  of  [£.  is  lis.  5d.  3  qr.  1 

37.  At  2c£.  5s.  per  cwt.,  what  cost  5  cwt.  3  qrs.  of  rai- 
sins ? 

Note.  2£.  5s.=2.25£.,  and  5  cwt.  3  qrs.=5.75  cwt. 
Multiplying  these  together,  the  result  is  12.9375^.  The 
decimal  part  of  this  result  may  be  changed  to  shillings  and 
pence  again.  .9375=£.  is  .9375  of  20  shillings  ;  therefore  if 
we  multiply  20  shillings  by  .9375,  or,  which  is  the  same 
thing,  if  we  multiply  .9375  by  20,  we  shall  obtain  the  answer 
in  shillings  and  parts  of  a  shilling.  This  is  evident  also 
from  another  course  of  reasoning.  .9375^.  is  now  in  pounds; 
if  it  be  multiplied  by  20  it  will  be  reduced  to  shillings. 
.9375 
20 


18.7500         The  result  is  18  shillings  and  .75  of  a  shil- 
ling, which  may  in  like  manner  be  reduced  to  pence  by  mul- 
tiplying it  by  12. 
.75 
12 

9.00        The  result  is  9d.     The  answer,  therefore, 
isl2<£.  18s.  9d. 

38.  What  cost  3  cwt.  2  qrs.  7  lb.  of  hops,  at  2^.  3s.  6d. 
per  cwt.  ? 

39.  What  cost  17  yds.  2  qrs.  2  nis.  of  broadcloth,  at  2£, 
5s.  7d.  per  yard  ? 

40.  What  cost  8  cwt.  1  qr.  13  lb.  of  wool,  at  3c£.  7s.  6d. 
per  cwt.  1 

41.  What  cost  3  hhds.  43  gals,  of  wine,  at  32=£.  14s.  8d. 
per  hhd.  ? 

8* 


90  ARITHMETIC.  PaH  I. 

42.  How  many  cwt.  of  raisins  in  1^  casks,  each  cask  conr 
taining  2  cwt.  0  qrs.  25  lbs  '' 

Note.  1}-1  6  and  2  cwt.  0.  qrs.  25  lb.i=2.2232-f  cwt. 
These  multnlied  together  produce  16.8957  cwt.  The  frac- 
tional part  ot  this  may  be  changed  to  quarters,  pounds,  &c. 
as  the  fractions  in  the  last  examples  were  changed  to  shil- 
lings and  pence.  .8957  cwt.  is  .8957  of  4  quarters,  or  it  is 
hundred-weights  and  may  be  reduced  to  quarters  and  pounds 
by  multiplying  by  4,  and  by  28. 
.8957 
4 


3.5828 
28 


The  result  is  3  qrs.  and  a  fraction. 

4G624  Then   multiply  .5828  qrs.  by  28,  it 

11656  gives    16    lb.    and   a   fraction   of   a 

pound.      Multiplying    .3184    lb.   by 

16.3184  16,  it  gives  5  oz.  and  a  fraction  of 

16  an  ounce. 


19104 
3184 


5.0944 
The  answer  Is  16  cwt.  3  qrs.  16  lb.  5yV  oz-  nearly.     The 
same  result  may  be  obtained  by  changing  the  decimal  .8957 
cwt.  to  a  common  fraction,  and  proceeding  according  to  the 
method  given  in  Art.  XVI. 

43.  How  many  cwt.  of  cotton  in  5f  bales,  each  bale  con- 
taining 4  cwt.  3  qrs.  7  lb.  ? 

44.  How  many  cwt.  of  coffee  in  13|  bags,  each  bag  con- 
taining I  cwt.  3  qrs.  15  lb.  1 

45.  Find  the  value  of  .387c£.  in  shillings,  pence,  and  far- 
things. 

46.  Find  the  value  of  .9842<£.  in  shillings,  pence,  and  far- 
things. 

47.  Find  the  value  of  .583  cwt.  in  quarters,  pounds,  &c. 

48.  Find  the  value  of  .23  cwt.  in  quarters,  pounds,  &e. 

49.  Find  the  value  of  .73648  cwt.  in  quarters,  pounds, 

50.  Find  the  value  of  .764s.  in  pence  and  farthings. 


XXVll.  DECIMALS.  91 

51.  Find  the  value  of  .3846  qr.    in  pounds  and  ounces. 

52.  Reduce  3.327  qrs.  to  pounds. 

53.  Reduce  4.6S4X.  to  pence. 

54.  Find  the  value  of  .340  of  a  day  in  hours,  minutes,  &c. 

55.  Find  the  value  of  .5870  of  an  hour  in  minutes  and 
seconds. 

56.  Express  in  decimals  of  a  foot  the  inches  in  the  follow- 
ing nambcrs  :— 3  ft.  G  in. ;  4  ft.  3  in. ;  7  ft.  9  in. ;  3  ft.  8 
in.;  5  ft.  7  in.;  9  ft.  10  in. 

57.  Find  the  value  of -375  ft.  in  inches  and  parts. 

58.  Find  the  value  of  .468  of  a  square  foot  in  square 
inches. 

59.  Find  the  value  of  .8438  of  a  solid  foot  in  solid  inches. 

60.  How  many  square  feet  in  a  board  9  in.  wide  and  15 
ft.  3  in.  long. 

Change  the  inches  to  decimals  of  a  foot.  Since  the  an- 
swer will  be  in  square  feet,  it  will  be  necessary  to  find  the 
value  of  the  decimal  in  square  inches.  In  general,  however, 
it  will  be  quite  as  convenient  to  let  the  answer  remain  in  de- 
cimals. The  answer  is  11.4375  ft.  It  will  be  sufficiently 
exact  to  call  it  11.4  ft. 

61.  How  many  square  feet  in  a  floor  14  ft.  7  in.  wide  and 
19  ft.  4  in.  long  ? 

62.  How  many  square  feet  in  a  board  1  ft.  8  in.  wide  and 
17  ft.  10  in.  long. 

63.  How  many  solid  feet  in  a  stick  of  timber  28  ft.  4  in. 
long.  1  ft.  2  in.  wide,  and  1 1  m.  deep  ? 

Note.  In  questions  of  this  kind  it  will  generally  be  most 
convenient  to  change  the  inches  to  decimals  of  a  foot,  be- 
cause when  the  whole  is  reduced  to  inches,  the  numbers  be- 
come very  large  and  the  operation  becomes  tedious.  Tenths, 
generally,  and  hundredths  in  almost  every  case,  will  be  suf- 
ficiently exact  for  common  purposes.  Those  who  measure 
timber,  boards,  wood,  &c.  would  find  it  extremely  convenient 
to  have  their  rules  divided  into  tenths  of  a  foot,  instead  of 
iinches. 

There  is  a  method  of  performing  examples  of  this  kind 
called  duodecimals,  which  will  be  explained  hereafter,  but  it 
is  not  so  convenient  as  decimals. 

64.  How  many  solid  feet  in  a  pile  of  wood  4  ft.  2  in.  wide, 
3  ft.  8  in.  high,  and  13  ft.  4  in.  long? 

It  has  been  already  remarked  that  in  interest,  discount. 


^  ARITHMETIC.  Part  1 

commi&sions,  &c.  6  per  cent,  7  per  cent.,  &c.  signifies  y^g- 
-j^^,  &.C.  of  the  sum.  This  may  be  written  as  a  decimal 
fraction.  In  fact  this  is  the  most  proper  and  the  most  con- 
venient way  to  express,  and  to  use  it.  1  per  cent,  is  .01  ; 
2  per  cent,  is  .02  ;  6  per  cent,  is  .06 ;  15  per  cent,  is  .15  ; 
6^  per  cent,  is  .065,  &c.  This  manner  of  expressing  the 
rate  will  be  very  simple  in  practice,  if  care  be  taken  to  point 
the  decimals  right  in  the  result. 

65.  A  commission  merchant   sold  a   quantity   of  goods 
amounting  to  $583.47,  for  which  he  was  to  receive  a  com- 
mission of  4  per  cent.     How  much  was  the  amount  of  the 
commission  1 
583.47 
.04 


823.3388  Ans, 
There  are  two  decimal  places  in  each  factor,  consequently 
there  must  be  four  places  in  the  result.     The  answer  is 
$23.34  nearly. 

(SQ.  What  is  the  commission  on  ^1358.27,  at  7  per  cent.  1 

67.  What  is  the  commission  on  $1783.425,  at  5  per  cent.  ? 

68.  A  merchant  bought  a  quantity  of  goods  for  $387.48, 
and  sold  them  so  as  to  gain  15  per  cent.  Ho-w  much  did  he 
ga'n,  and  for  how  much  did  he  sell  the  goods  ? 

69.  What  IS  the  insurance  of  a  ship  and  cargo,  worth 
$53250,  at  ^  per  cent.  ? 

Note.  2i  per  cent  is  equal  to  .025,  for  2  per  cent,  is  .02, 
and  ^  per  cent,  is  ^  of  an  hundredth,  which  is  5  thousandths. 

70.  What  is  the  duty  on  a  quantity  of  books,  of  which  the 
invoice  is  $157.37,  at  15  per  cent.  1 

Note.  It  is  usual  at  the  custom-house  to  add  y^  or  10 
per  cent,  to  the  invoice  before  casting  the  duties.  10  per 
cent,  on  $157.37  is  $15,737,  which,  added  to  $157.37 
makes  $173,107.  The  duties  must  be  reckoned  on  $173,107. 
When  the  duties  are  stated  at  15  per  cent,  they  will  actually 
be  16^-  per  cent,  on  the  invoice  ;  because  15  per  cent,  on 
^  will  amount  to  1^  per  cent,  on  the  whole.  14:  will  be 
most  convenient  generally  to  reckon  the  duties  at  16^  per 
cent.,  instead  of  adding  y'^  of  the  sum  and  then  reckoning 
them  at  15  per  cent.  When  the  duties  are  at  any  other  rate, 
the  rate  may  be  increased  -^^  of  itself,  instead  of  increasing 


XXVIT.  DECIMALS.  93 

the  invoice  -j\.  For  instance,  iftlie  rate  is  10  per  cent,  call 
it  11  per  cent.,  if  the  rate  is  14  per  cent,  call  it  lo/^-  per 
cent.,  then  the  multiplier  will  be  .154.  If  the  rate  is  12^ 
percent.,  that  is,  .1'25,  -J^  of  this  is  .0125,  which  added  to 
.125  makes  .1375  for  the  multiplier. 

71.  What  is  the  duty  on  a  quantity  of  tea,  of  which  the 
mvoice  is  $215.17,  at  50  per  cent.  ? 

72.  What  is  the  duty  on  a  quantity  of  wine,  of  which  the 
invoice  is  8873,  at  40  per  cent.  1 

73.  What  is  the  duty  on  a  quantity  of  saltpetre,  of  which 
the  invoice  is  $1157,  at  7i  per  cent.  ? 

74.  Imported  a  quantity  of  hemp,  the  invoice  of  which 
was  $1850,  the  duties  13|  per  cent.  What  did  the  hemp 
amount  to  after  the  duties  were  paid  1 

75.  Bought  a  quantity  of  goods  for  858.43,  but  for  cash 
the  seller  made  a  discount  of  20  per  cent.  What  did  the 
goods  amount  to  after  the  discount  was  made  ? 

76.  A  merchant  bought  a  quantity  of  sugar  for  $183.58, 
but  being  damaged  he  sold  it  so  as  to  lose  7|-  per  cent. 
How  much  did  he  sell  it  for  ? 

77.  Bought  a  book  for  $.75,  but  for  cash  a  discount  of 
20  per  cent,  was  made.     What  did  the  book  cost  1 

78.  Bought  a  book  for  $4,375,  but  for  cash  a  discount 
of  15  per  cent,  was  made.     How  much  did  the  book  cost  I 

79.  What  is  the  interest  of  $43.25  for  1  year,  at  6  per 
cent.  1 

80.  What  is  the  interest  of  $183.58  for  1  year  at  7  per 
:ient.  1 

81.  At  6  per  cent,  for  1  year,  Avhat  would  be  the  rate  per 
cent,  for  2  years  1    For  3  years  ?    For  4  years  1 

82.  At  6  per  cent,  for  1  year,  what  would  be  the  rate  per 
tent,  for  6  months  1  For  2  months  i  For  4  months  1  For 
1  month  1  For  3  months  ?  For  5  months  1  F(.«r  7  months  ? 
For  8  months  ?  For  9  months  ?  For  10  months  ?  For  11 
months  1 

83.  At  6  per  cent,  for  1  year,  what  would  be  the  rate  per 
cent,  for  13  months?  For  14  months?  For  1  year  and  5 
months  ? 

84.  If  the  rate  for  60  days  is  1  per  cent.,  or  .01,  what  is 
the  rate  for  6  days?  For  12  days?  For  18  days?  For 
24  days  ?  For  36  days  ?  For  42  days  ?  For  48  days  ?  For 
54  days  '^ 


94  ARITHMETIC.  Part  1. 

Note.  The  interest  of  6  days  is  Jg-  per  cent.,  that  is  .001. 
The  interest  of  I  day  therefore  will  be  \  of  y^g-,  or  ^^  per 
cent.,  or  .00016.  The  rate  for  2  days  twice  as  much,  &c. 
In  fact  the  rate  for  the  days  may  always  be  found  by  divid- 
ing the  number  of  days  by  6,  annexing  zeros  if  necessary, 
and  placing  the  first  figure  in  the  place  of  thousandths,  if 
the  number  of  days  exceeds  6. 

85.  What  is  the  interest  of  $47.23  for  2  months,  at  6  per 
cent.  ? 

Note,  When  the  rate  per  cent,  is  stated  without  men- 
tioning the  time,  it  is  to  be  understood  for  1  year,  as  in  the 
followmg  examples. 

86.  What  is  the  interest  of  $27.19  for  4  months,  at  6  per 
cent.  1 

87.  What  is  the  interest  of  $147.96  for  6  months,  at  6 
per  cent.  1 

88.  What  is  the  interest  of  $87,875  for  8  months,  at  G 
per  cent.  ? 

89.  What  is  the  interest  of  $243.23  for  14  months,  at  6 
per  cent.  1 

90.  What  is  the  interest  of  $284.85  for  3  months,  at  6 
per  cent.  1 

91.  What  is  the  interest  of  $28.14  for  5  months,  at  6  per 
cent.  1 

92.  What  is  the  interest  of  $12.18  for  7  months,  at  6  per 
cent.  1 

93.  What  is  the  interest  of  $4.38  for  9  months,  at  6  per 
cent.  1 

94.  What  is  the  interest  of  $15,125  for  II  months,  at  6 
per  cent.  1 

95.  What  is  the  interest  of  $127.47  for  2  months  and  12 
days,  at  6  per  cent.  ? 

96.  What  IS  the  interest  of  $873,62  for  4  months  and  24 
days,  at  6  per  cent.  ? 

97.  What  is  the  interest  of  $115.42  for  7  months  and  15 
days,  at  6  per  cent.  1 

98.  What  is  the  interest  of  $516.20  for  11  months  and 
23  days,  at  6  per  cent.  ? 

99.  What  is  the  interest  of  $143.18  for  1  year,  7  months, 
and  14  day?,  at  6  per  cent.  ? 

100.  A  gave  B  a  note  for  $357.68  on  the  13th  Nov. 
1819,   and  paid  it  on   the  11th  April,    1822,  interest  at  6 


XXVa.  DECIMALS.  95 

per  cent.     How  much   was   the  principal  and   interest  to- 
gether at  the  time  of  payment  ? 

101.  A  note  for  $84;5.43  was  given  5th  July,  1817,  and 
paid  14th  April,  1822,  interest  at  0  per  cent.  How  much 
did  the  principal  and  interest  amount  to  ? 

102.  A  note  was  given  7th  March,  1818,  for  $587;  a 
payment  was  made  19th  May,  1819,  of  $53,  and  the  rest 
was  paid  11th  Jan.  1820.  What  was  the  interest  on  the 
note  ? 

103.  What  is  the  interest  of  $157  for  2  years,  at  5  per 
cent.  ? 

104.  What  is  the  interest  of  13c£.  3s.  6d.  for  1  year,  at 
6  per  cent.  ? 

Note.  If  the  shillings  be  reduced  to  a  decimal  of  a  pound, 
the  operation  will  be  as  simple  as  on  Federal  money.  The 
following  is  a  more  simple  method  of,  changing  shillings  to 
decimals,  than  the  one  given  above.  ^V  P^*"^  ^^^  ^^^:  '^  ^^"' 
therefore  every  2s.  is  ^\£.  or  A£.  Every  shilling  is  /q-c^., 
that  is  T-f^i:.  or  .05i:.  ;   3s.  then  is  .\£.  and  .05^'.,  or  .15^. 

In  lc£.  there  are  960  farthings.  1  farthing  then  is  -p-^o  of 
\£.  (jd.  is  24  farthings,  consequently  ^Yo  "^  ^  ^'  '^^^^® 
are  rather  larger  than  thousandths,  but  they  are  so  near 
thousandths  that  in  small  numbers  they  may  be  used  as  thou- 
sandths. -§^^-^£.z^^-^£.  when  reduced,  and  ^|f^=£.=Vo^., 
so  that  24  farthings  are  exactly  -^lU^-  or  .025^'.  If  the 
number  of  farthings  is  13  they  will  be  rofo^-  and  rather 
more  than  \  of  another  thousandth.  This  may  be  called 
_i  4_  or  .014,  and  the  error  will  be  less  than  4-  of  j-^^^.  If 
the  number  of  farthings  be  less  than  12  they  may  be  called 
so  many  thousandths,  and  the  error  will  be  less  th^n  1  of 
^_i__.  If  the  number  of  farthings  is  between  12  and  30  add 
1  to  them,  if  more  than  30  add  2,  and  call  them  so  many 
thousandths  ;  and  the  result  will  be  correct  within  less  than 
I  of  j-.^oo^.  48  farthings  make  1  shilling,  therefore  there 
will  never  be  occasion  to  use  more  than  this  number.  From 
the  above  observations  we  obtain  the  following  rule.  Call 
every  two  shtlling^s  one  tenth  of  a  pound,  every  odd  shilling 
five  hundredths,  and  the  numher  of  farthings  in  the  pence 
and  farthings  so  many  thousandths,  adding  one  if  the  num- 
her is  between  twelve  and  thirty-six,  and  two  if  more  than 
thirty-six. 

It  will  be  well  to  rememb(;r  this  rule,  because  it  will  be 


96  ARITHMETIC.  Part  1. 

useful  in  many  instances,  particularly  in  changing  English 
money  to  dollars  and  cents,  and  the  contrary. 

13c£.  3s.  6d.  then  is  reduced  as  follows  :  2s. =.1^".  ls.r= 
.05£.  and  6d.izr24  farthings:rr.025^.  and  the  whole  is  equal 
to  .£13.175. 

13.175 
.06 


£  .79050  Ans. 
To  change  the  result  to  shillings  and  pence  it  is  necessary 
to  reverse  the  above  operation.  The  .7  or  -^^  are  14s.  The 
.09  or  ^f  0  are  ^|^-f  ^4._.  xhe  ^^  are  Is.  and  the  ^^  are 
^l-Q,  or  40  farthings  ;  then  taking  out  2,  because  the  num- 
ber is  above  36,  we  have  38  farthings,  or  9d.  2qr.  ;  and  the 
whole  interest  is  15s.  9d.  2qr. 

105.  What  is  the  interest  of  13<£.  15s.  3d.  2  qr.  for  1  year 
and  6  months,  at  6  per  cent.  ? 

106.  What  is  the  interest  of  4c£.  lis.  8d.  Iqr.  for  9  months 
and  15  days,  at  6  per  cent.  T 

107.  What  is  the  interest  of  137^.  Os.  9d.  from  13th  May. 
1811,  to  19th  July,  1815,  at  6  per  cent.  ? 

108.  What  is  the  interest  of  137i:.   17s.  2d.   from   11th 
Jan.  1822,  to  15th  August,  at  6  per  cent.  1 

109.  What  is    the   interest  of  \1£.  9s.   from  1st  June, 
1819,  to  17th  Aug.  1820,  at  6  per  cent.  ? 

110.  What  is  the  interest  of  13s.  4d.  from  17th  June, 
1818,  to  2Sth  Aug.  1821,  at  6  per  cent.  ? 

1 1 1.  What  is  the  interest  of  4s.  8d.  2qr.  for  7  months  and 
3  days,  at  6  per  cent  ? 

1 12.  What  is  the  commission  on  143c£.  13s.,  at  5  per  cent.  1 

113.  What  is  the  duty  on  a  quantity  of  goods,  of  which 
tlie  invoice  is  2>7c£.  19s.  4d.,  at  15  per  cent.  1 

N.  B.     The  above  examples  in  pounds,  shillings,  &c.  ap- 
ply equally  to  English  and  to  American  money. 


Division  of  Decimals, 

XXVIII.  1.  If  5  barrels  of  cider  cost  $18.75,  what  js 
that  per  barrel  1 

2.  A  man  bought  17  sheep  for  $98.29,  what  was  the  ave- 
rage price  ? 


XXVIII.  DECIMALS.  97 

3.  Divide  $183,575  equally  among  5  men.     How  much 
will  each  have  1 

4.  Divide  7.5  barrels  of  flour  equally  among  5  men.    How 
much  will  they  have  apiece  ? 

5.  Divide   11.25  bushels  of  corn  equally  among  8  men. 
How  much  will  they  have  apiece  ? 

6.  A  man  travelled  73.487  miles  in  15  hours  ;  what  was 
the  average  distance  per  hour  1 

7.  At  ^S£.  5s.  8d.  per  ton,  what  cost  1  cwt.  of  iron  1 

S.  If  a  ship  and  cargo  are  worth  1253£.  6s.  4d.,  what  is 
the  man's  share  who  owns  ,'5  of  her  ? 
9.  Whatis  1  of49.376? 

10.  What  is -Jy  of  583..542  ? 

11.  What  is -J-^  of  13.75 '? 
12    Wliat  is  ;-i^  of  387.65  1 

13.  Divide  13.8468  by  4. 

14.  Divide  1387.35  by  48. 

15.  Divide  158.6304  by  113. 

16.  Divide  12.4683  by  27. 

17.  Divide  1.384  by  1.5. 

18.  Divide  .7376  by  28. 

19.  Divide  .6438  by  156. 

20.  Divide  1.5  by  58. 

21.  Divide  .4  by  13. 

22.  Divide  .0346  by  27. 

23.  Divide  .003  by  43. 

24.  Divide  1.06438  by  1846. 

25.  Divide  13.84783  by  137648. 

26.  At  $1.37  per  gallon,  how  many  gallons  of  wine  may  be 
bought  for  837  ? 

27.  At  8-34  per  bushel,  how  many  bushels  of  oats  may 
be  bouffht  for  $24  ? 

29.  At  8-165  per  lb.,  how  many  lb.  of  raisins  may  be 
bought  for  83  ? 

30.  At  8.03  apiece,  how  many  lemons  may  be  bought  for  $5 1 

31.  If  1.75  yards  of  cloth  will  make  a  coat,  how  many 
coats  may  be  made  from  38  yards  1 

32.  If  1.3  bushels  of  rye  is  sufticient  to  sow  an  acre  of 
ground,  how  many  acres  will  23  bushels  sow  1 

33.  If  18.75  bushels    of   wheat   grow  on   1    acre,   how 
many  acres  will  produce  198  bushels,  at  that  rate  1 

34.  If  a  man  travel  5.3S5  miles  in  an  hour,  in  how  many 
hours  will  he  travel  S3  miles  at  that  rate  ? 

9 


98  ARITHMETIC.  Part  1. 

35.  If  3s.    will  pay  for  1  day's  work,  how  many  days' 
work  may  be  had  for  13s.  ? 

36.  If  5s.  8d.  will  pay  for  1  day's  work,  how  many  days' 
work  will  llc£.  pay  for  1 

37.  At  8s.  3d.  per  gallon,  how  many  gallons  of  wine  may 
be  bought  for  18c£.  ? 

38.  If  2.5  barrels  of  cider  cost  $7,  what  is  that  per  bar- 
rel? 

39.  If  1.5  barrel  of  flour  cost  $10,  what  is  that  per  bar- 
rel? 

40.  If  2.75  firkins  of  butter  cost  $23,  what  is  that  per 
firkin  ? 

41.  If  3.375  barrels  of  beer  cost  $14,  what  is  that  per 
barrel  ? 

42.  If  13.16  bushels  of  wheat  cost  6^.,  what  is  that  per 
bushel  ? 

43.  If  .8  of  a  yard  of  cloth  cost  $2,  what  is  that  per  yard  \ 

44.  If  .35  of  a  ton  of  hay  cost  $8,  what  cost  a  ton  ? 

45.  If  .846  of  a  barrel  of  flour  cost  32  shillings,  what 
will  a  barrel  cost  at  that  rate  ? 

46.  If  .137  of  a  ton  of  iron  cost  52  shillings,  what  will  1 
ton  cost  1 

47.  How  many  times  is  1.3  contained  in  18  ? 

48.  How  many  times  is  3.25  contained  in  39  ? 

49.  How  many  times  is  4.75  contained  in  180  ? 

50.  How  many  times  is  16.375  contained  in  4,876  1 

51.  How  many  times  is  24.538  contained  in  63  ? 

52.  How  many  times  is  1.372  contained  in  14  1 

53.  How  many  times  is  4.1357  contained  in  15  1 

54.  How  many  times  is  .3  contained  in  3  ? 

55.  How  many  times  is  .04  contained  in  4  ? 

56.  How  many  times  is  .13  contained  in  8  1 

57.  How  many  times  is  .385  contained  in  17 1 

58.  How  many  times  is  .0684  contained  in  47  ? 

59.  How  many  times  is  .0001  contained  in  53  ? 

60.  How  many  times  is  .0005  contained  in  127  1 

61.  3  is  .3  of  what  number  ? 

62.  4  is  .04  of  what  number  1 

63.  8  is  .13  of  what  number  ? 

64.  17  is  .385  of  what  number  1 

65.  47  is  .0684  of  what  number  ? 

66.  53  is  .0001  of  what  number  1 

67.  127  is  .0005  of  what  number? 


XXVIII.  DECIMALS.  90 

G8.  How  many  times  is  .0035  contained  in  67  1 

69.  67  is  .0035  of  what  number  ] 

70.  Divide  156  by  4.35. 

71.  Divide  38  by  13.56. 

72.  Divide  23  by  1.3846. 

73.  Divide  7  by  8.4. 

74.  7  is  what  part  of  8.4  1 

75.  Divide  3  by  5.8. 

76.  3  is  what  part  of  5.8  ? 

77.  Divide  8  by  17.37. 

78.  8  is  what  part  of  17.37  1 

79.  Divide  23  by  120.684. 

80.  23  is  what  part  of  120.684  1 

81.  Divide  14  by  .7. 

82.  Divide  130  by  .83. 

83.  Divide  847  by  .134. 

84.  Divide  8  by  .0645. 

85.  Divide  3  by  .00735. 

86.  Divide  1  by  .005643. 

87.  Divide  157  by  .00001. 

88.  At  $2.75  per  gallon,  how  many  gallons  of  wine  may 
be  bought  for  856.03  ? 

89.  At  17.375  shillings  per  gallon,  how  many  gallons  of 
wine  may  be  bought  for  42.25  shillings  ? 

90.  At  16s.  4d.  per  gallon,  how  many  gallons  of  brandy 
may  be  bought  for  4=£.  7s.  1 

91.  At  2£.  3s.  4d.  per  barrel,  how  many  barrels  of  flour 
may  be  Ixjught  for  32^'.  7s.  6d.  ? 

92.  If  3.75  barrels  of  flour  cost  $25.37,  how  much  is  that 
per  barrel  ? 

93.  If  5.375  barrels  of  cider  cost  4^.  4s.,  what  is  that  per 
barrel  ? 

94.  If  .845  of  a  yard  of  cloth  cost  $5.37,  what  is  that  per 
yard  ? 

95.  If  4  of  a  ton  of  iron  cost  860.45,  what  cost  1  ton  1 
9f).  How  many  times  is  13.753  contained  in  42.7  1 

97.  How  many  times  is  1.468  contained  in  473.75  1 

98.  How  many  times  is  .7647  contained  in  13.42  1 

99.  How  many  times  is  .0738  contamed  in  1.6473  1 

100.  1.6473  is  .0738  of  what  number  ? 

101.  How  many  times  is  .001  contained  in  .1  1 

102.  .1  is  .001  of  what  number  1 

103.  How  many  times  is  .002  contained  in  .01  1 


100  ARITHMETIC.  Part  1. 

104.  .01  is  .002  of  what  number  1 

105.  How  many  times  is  .002  contained  in  .002  \ 

106.  .002  is  .002  of  what  number  ? 

107.  Divide  31.643  by  2.3846. 

108.  Divide  2.4637  by  .6847. 

109.  If  1  lb.  of  candles  cost  8.14.  how  many  lb.  may  be 
bought  for  81.375  1 

110.  If  4.5  yards  of  cloth  cost  828.35,  how  much  is  that 
per  yard  ? 

111.  If  3.45  tons  of  hay  cost  22£.  7s.  5d.,  how  much  is 
tliat  per  ton  ? 

112.  At  3s.  8d.  per  bushel,  how  many  bushels  of  barley 
may  be  bought  for  3^.  5s.  7d.  1 

113.  If  47.25  bushels  of  barley  cost  15£.  17s.  5d.,  what 
is  that  per  bushel  ? 

114.  If  15  cwt.  3  qr.  14  lb.  of  iron  cost  17.£.  14s.  8d., 
what  is  that  per  cwt.  1 

115.  If  .35  of  a  ton  of  iron  cost  10£.  .3s.  5d.,  what  cost  a 
ton  at  that  rate  ? 

116.  Divide  16.4567  by  2.5. 

116.  Divide  137.06435  by  3.25. 

117.  Divide  105.738  by  .3. 

118.  Divide  75.426  by  .1. 

119.  Divide  1.76453  by  1.3758. 

120.  Divide  .78357  by  .001. 

121.  Divide  .073467  by  .005. 

122.  Divide  .007468  by  .0075. 

123.  How  many  times  is  .037  contained  in  1.04738? 

124.  1.04738  is  .037  of  what  number  ? 

125.  How  many  times  is  .135  contained  in  13.4073  1 

126.  13.4073  is  .135  of  what  number  ? 

127.  Divide  13.40764  by  123.725. 

128.  Divide  .406478  by  135.407. 

In  the  following  examples  express  the  division  in  the  form 
of  a  common  fraction,  and  reduce  them  to  their  lowest  terms. 

129.  Divide  17.57  by  14.23. 

130.  Divide  3.756  by  5.873. 

131.  Divide  .6375  by  .5268. 

132.  Divide  3.45  by  2.756. 

133.  Divide  1.6487  by  2.35. 

134.  Divide  113.45  by  21.4764. 

135.  Divide  .7384  by  .37. 


XXVIII.  DECIMALS.  101 

136  Divide  .007  by  .5. 

137.  Divide  .047387  by  .0042. 

138.  Divide  .53  by  .00067. 

139.  Divide  .003  by  0.00001. 

140.  3.5  is  what  part  of  7.8  ? 

141.  13.70  is  what  part  of  17.5  ? 

142.  7.0387  is  what  part  of  42.95  ? 

143.  1.5064  is  what  part  of  8.944783  ? 


Miscellaneous  Examples. 

1.  If  1.4  cwt.  of  sugar  cost  $10.09,  what  wdl  9  cwt.  3 
qrs.  cost  ? 

2.  If  19|  yards  of  cloth  cost  $128.35,  what  will  18  yds.  3 
qrs.  cost  1 

3.  If  23|  yds.  of  riband  cost  $5|,  what  will  34f  yds.  cost  ? 

4.  If  3  cwt.  2  qrs.  14  lb.  of  sugar  cost  $38.55  what  will 
19  cwt.  1  qr.  17  lb.  cost  ? 

5.  If  i  cwt.  of  tobacco  cost  4^.  18s.,  how  much  may  be 
bought  for  I3i:.  17s.  8d.  ? 

6.  Sold  75f  chaldrons  of  lime,  at  lis.  Gd.  per  chaldron  , 
how  much  did  it  come  to  ? 

7.  A  goldsmith  sold  a  tankard  for  W£.  13s.,  at  the  rate 
of  5s.  6d.  per  oz. ;  how  much  did  it  weigh? 

8.  Goliah  the  Philistine  is  said  to  have  been  6|  cubits 
high,  each  cubit  being  1  ft.  7.168  English  inches  ;  what  was 
his  height  in  English  feet  ? 

9.  How  many  yards  of  flannel  that  is  1  English  ell  wide 
will  be  sufficient  to  line  a  cloak  containing  8^  yds.,  that  is  ^ 
yd.  wide  1 

10.  I  agreed  for  a  carriage  of  2.5  tons  of  goods  2.9  miles, 
for    .75  of  a  guinea  ;  what  is  that  per  cwt.  for  J  mile  1 

11.  If  a  traveller  perform  a  journey  in  35.3  days,  when 
the  days  are  11.374  hours  long  ;  in  how  many  days  will  he 
perform  it,  when  the  days  are  9.13  hours  long? 

12.  If  12  men  can  do  125  rods  of  ditching  in  65|  days; 
in  how  many  days  can  they  do  242 ^^^  rods  ? 

13.  In  a  room  18  ft.  0  in.  long,  and  14  ft.  9  in.  wide,  how 
many  square  feet  ?  In  a  yard  of  carpeting  that  is  2  ft.  8  in. 
wide,  how  many  square  feet  ?  How  many  yards  of  such  car- 
peting will  cover  the  above  mentioned  floor  ? 

9* 


102  ARITHMETIC.  Part  \. 

14.  How  many  yards  of  carpeting  that  is  \\  yd.  wide  will 
cover  a  floor  22  ft.  7  in.  long,  and  19  ft.  8  in.  wide  ? 

15.  How  many  feet  of  boards  will  it  take  to  cover  the 
walls  of  a  house  32  ft.  S  in.  long,  26  ft.  4  in.  wide,  and  26 
ft.  5  in.  high  ?  How  much  will  they  cost  at  $3.50  per  1000 
feet? 

16.  How  many  feet  will  it  take  to  cover  the  floors  of  the 
above  house  1 

17.  If  1000,  or  a  bunch,  of  shingles  will  cover  10  feet 
square,  how  m„any  bunches  will  it  take  to  cover  the  roof  of 
the  above  house,  allowing  the  length  of  the  rafters  to  be  IG 
ft.  5  in.  1 

18.  In  a  piece  of  land  37f  rods  long,  and  32f  rods  wide, 
how  many  acres  ? 

19.  What  will  a  piece  of  land,  measuring  57  ft.  in  length, 
and  43  ft.  in  breadth,  come  to,  at  the  rate  of  $25  per  square 
rod? 

20.  In  a  pile  of  wood  23  ft.  7  in.  long,  3  ft.  10  in.  wide, 
and  4  ft.  3  in.  high,  how  many  cords  ? 

21.  How  many  feet  of  wood  in  a  load  S  ft.  long,  4  ft. 
wide,  and  3  ft.  8  in,  high  ? 

N.  B.  Wood  prepared  for  the  market  is  generally  4  feet 
long,  and  a  load  in  a  wagon  generally  contains  two  lengths, 
or  8  feet  in  length.  If  a  load  is  4  feet  high  and  4  feet  wide, 
it  contains  a  cord.  It  was  reniarked  above,  that  what  is 
called  one  foot  of  vvood,  is  16  solid  feet,  and  that  8  such  feet 
make  1  cord.  To  find  how  many  of  these  feet  a  pile  or  load 
of  wood  contains,  it  is  necessary  to  find  the  number  of  solid 
feet  in  it,  and  then  to  divide  by  16.  W^hen  the  load  of  wood 
IS  8  feet  long,  we  may  multiply  the  breadth  and  height  to- 
gether, and  then,  instead  of  multiplying  by  8,  and  dividing 
by  16,  we  may  divide  at  first  by  2,  and  the  same  result  will 
be  obtained. 

22.  How  many  feet  of  wood  in  a  load  8  feet  long,  3  ft.  4 
in.  wide,  and  2  ft.  7  in.  high  ? 

23.  How  many  feet  of  wood  in  a  load  8  feet  long,  3  ft.  7 
in.  wide,  and  5  ft.  2  in.  high  1 

24.  How  much  wood  in  a  load  8  ft.  long,  4  ft.  2  in.  wide, 
and  5  ft.  4  in.  high  1 

25.  If  a  load  of  wood  is  8  ft.  long,  and  3  ft.  7  in.  wide, 
how  high  must  it  be  to  make  a  cord  ? 

26.  How  manj  bricks  8  inches  long,  4  inches  wide,  and 


XXVIII.  DECIMALS.  103 

2]  inches  thick,  will  it  take  to  build  a  house  44  feet  long, 
40  feet  wide,:20  feet  high,  and  the  walls  12  inches  thick  1 

27.  What  is  the  value  of  87  pigs  of  lead,  each  weighing  3 
cwt.  2  qrs.  17^  lb.,  at  8£.  13s.  8d.  per  fother  of  19^  cwt.  1 

28.  What  is  the  tax  upon  $1153.  at  $.03  on  a  dollar  1 

29.  What  is  the  tax  upon  $843.35,  at  $.04  on  a  dollar  ? 

30.  What  is  the  tax  upon  785i:.  lis.  4d.  at2s.5d.  on  the 
pound  ? 

31.  Suppose  a  certain  town  is  to  pay  a  tax  of  $614.5.88, 
and  the  whole  property  of  the  town  is  valued  at  $153647  ; 
what  is  that  on  a  dollar  ?  How  much  must  a  man  pay, 
whose  property  is  valued  at  $23475.67 1 

Note.  In  assessing  taxes,  the  first  requisite  is  to  have  an 
inventory  of  ihe  property,  both  real  and  personal,  of  the  whole 
town  or  parish,  and  also  of  each  individual  who  is  to  be  tax- 
ed, and  the  number  of  polls.  The  polls  are  always  stated  at 
a  certain  rate.  Then  knowing  the  whole  tax,  take  out  what 
the  polls  amount  to,  and  the  remainder  is  to  be  laid  upon  the 
property.  Find  how  much  each  dollar  is  to  pay,  and  make 
a  table,  containing  the  portion  for  1,  2,  3,  &c.  to  10  dollars, 
Aen  for  20,  30,  40,  &c.  to  100,  and  then  for  200,  300,  &c. 
From  this  table  it  will  be  easy  to  fixid  the  tax  upon  the  pro- 
perty of  any  individual. 

32.  A  certain  town  is  taxed  $3137.43.  The  whole  pro- 
perty of  the  town  is  valued  at  $89640.76.  There  are  120 
polls  which  are  taxed  $.75  each.  What  is  the  tax  on  ^  dol- 
lar 1  How  much  is  a  man's  tax  who  pays  for  3  polls,  and 
whose  property  is  valued  at  $2507  1 

33.  A  merchant  bought  wine  for  $1.75  per  gallon,  and 
sold  it  for  $2.25  per  gallon.     What  per  cent,  did  he  gain  1 

Note.  He  gained  50  cents  on  a  gallon,  which  is  tVs^M 
ofthe  first  cost.  It  has  been  already  remarked  that  1  per  cent, 
is  .01,  2  per  cent,  is  .02,  &c.  ;  that  is,  the  rate  per  cent,  is 
always  a  decimal  fraction  carried  to  two  places  or  hundredths. 
To  find  the  rate  per  cent,  then,  first  make  a  common  frac 
tion,  and  then  change  it  to  a  decimal  if =.285.  Now  .28 
is  28  pel  cent,  and  .0055  isf  ^^^jper  cent.  The  rate  then  28yV„ 
per  cent.  The  two  first  decimal  places  taken  together  be- 
mg  hundredths  are  so  much  per  ceiit.,  and  thousandths  are 
so  many  tenths  of  one  per  cent. 

34.  A  merchant  bought  a  hhd.  of  molasses  for  $20,  and 
sold  it  for  $25 ;  what  per  cent,  did  he  gain  1 


104  ARITHMETIC.  Part  I. 

35.  A  merchant  bought  a  quantity  of  flour  for  8137,  and 
sold  it  for  $143  ;  what  per  cent,  did  he  gain  ? 

36.  A  man  bouglit  a  quantity  of  goods  for  $94.37,  and 
sold  them  for  $83.9'2.     What  did  he  lose  per  cent.  '! 

37.  A  merchant  bought  molasses  for  Is.  8d.  per  gallon, 
and  sold  it  for  2s.  3d.  per  gallon.    What  did  he  jain  per  cent  ? 

38.  A  merchant  bought  wine  for  lis.  3d.  per  gallon,  and 
sold  it  for  9s.  S.^d.     What  per  cent,  did  he  lose  ? 

39.  A  merchant  bought  a  quantity  of  goods  for  37<£.  15s 
8d. ;  and  sold  them  again  for  43^.  lis.  4d.  What  per  cent 
did  he  gain  ? 

40.  A  man  buys  a  quantity  of  goods  for  $843  ;  what  per 
cent,  profit  must  he  make  in  order  to  gain  $157  ? 

41.  A  man  failing  in  trade  owes  $19137.43,  and  his  pro- 
perty is  valued  at  $13472.19.     What  per  cent,  can  he  pay  ? 

42.  A  man  purchased  a  quantity  of  goods,  the  price  of 
which  was  $57,  but  a  discount  being  made,  he  paid  $45.60. 
What  per  cent,  was  the  discount  ? 

43.  A  man  hired  $37  for  1  year,  and  then  paid  for  princi- 
pal and  interest  $92.22.  What  was  the  rate  of  ihe  in- 
terest ? 

44.  A  man  paid  $12.81  interest  for  $183,  for  2  years. 
What  was  me  rate  per  year  ? 

45.  A  man  paid  $13,125  interest  for  $135,  for  1  year  and 
6  months.     What  was  the  rate  per  year  1 

46.  A  man  paid  $4.37  interest  for  $58,  for  1  year  and  8 
months.     What  was  the  rate  per  year  ? 

47.  4s.  6d.  sterling  of  England  is  equal  to  1  dollar  in  the 
United  States.  What  is  the  value  of  \£.  sterling  in  Federal 
money  ? 

48.  How  many  dollars  in  35c£.  sterling  ? 

49.  How  many  dollars  in  27^.  14s.  8d.  1 

Note.  Change  the  shillings  and  pence  to  the  decimal  of 
a  pound,  by  the  short  method  shown  above. 

50.  How  many  dollars  in  187c£.  17s.  4d.  1 

51.  In  $19.42  how  many  pounds  sterling  ? 

52.  In  $157  how  many  pounds  ? 

53.  In  $2334.72  how  many  pounds  ? 

54.  Bought  goods  in  England  to  the  amount  of  123.£.  17s. 
9d. ;  expenses  for  getting  on  board  3^\  5s.  8d.  ;  $8.50 
freight;  duties  in  Boston  15  per  cent,  on  the  invoice  ;  other 
expenses  in  Boston  $15.75.  How  many  dollars  did  the 
goods  cost  1  How  much  must  they  be  sold  for  to  gain  12 
per  cent,  on  the  cost  1 


XXVIII.  DECIMALS.  105 

55.  What  is  the  interest  of  $47,50  for  1  year,   7  months., 
and  13  days,  at  7  per  cent.  ? 
47.50 
.07 


3.3250     Interest  for  I  year. 
].()G'25         do.       for  (>  months. 

.277-f-       do.       for  1  montli. 

.092-1-       do.       for  10  days. 

.03  nearly  do.      for  3  days. 


Ajis.  5.3865 

I  first  find  the  interest  for  1  yeai,  anc  then  i^  of  that  is  the 
interest  for  6  months  ;  -]  of  the  interest  for  6  months  will  be 
the  interest  for  1  month  ;  -^  of  the  interest  for  1  month  wilJ 
be  the  interest  for  10  days,  and  -i  of  the  interest  for  10  days 
is  very  near  the  interest  for  3  days.  All  these  added  to- 
gether will  give  the  interest  for  the  whole  time.  In  a  simi- 
lar manner,  the  interest  for  any  time  at  any  rate  per  cent, 
may  be  calculated. 

When  there  are  months  and  days,  it  is  better  to  calculate 
the  interest  first  at  G  or  12  per  cent.,  and  then  change  it  to 
the  rate  required.  Observe  that  1  per  cent,  is  |-  of  (i  per 
cent.,  H  per  cent,  is  J  of  6  per  cent.,  2  per  cent  is  ^  of  6 
per  cent,  &c.  Hence  if  the  rate  is  7  per  cent.,  calculate 
first  at  6  per  cent.,  and  then  add  |  of  it  to  itself,  or  if  5  per 
cent.,  subtract  -}  ;  if  7-|  or  4|  per  cent,  add  or  subtract  ^,  &c. 

Let  us  take  the  above  example. 

6  per  cent,  for  1  year,  7  months,  and  13  days,  is  9/^  per 
cent,  nearly,  that  is  .097. 
47.50 
.097 


33250 
42750 


^  of  4.60750  Interest  at  6  per  cent. 
7679        do.      at  1  per  cent. 


85.3754 

This  answer  agrees  with  the  other  within  about  1  cent. 
Greater  accuracy  might  be  attained,  by  carrying  the  rate  to 
one  or  two  more  decimal  places. 


106  ARITHMETIC.  Part  1. 

56.  What  is  the  interest  of  $135.16  from  the  4th  June, 
1817  to  13th  April,  1818,  at  5  per  cent.  ? 

57.  What  is  the  interest  of  885.37  from  13th  July,  1S15, 
to  17th  Nov.  1818,  at  4^  per  cent.  ? 

58.  What  is  the  interest  of  845.87  from  19th  Sept.  1810, 
to  lUh  Aug.  1821,  at  7^  per  cent.  ? 

59.  What  is  the  interest  of  $183  from  23d  Oct.  1817,  to 
19th  Jan.  1820,  at  4  per  cent.  ? 

60.  ^^^hat  is  the  interest  of  li3=€.  14s.  for  1  year,  5  monihs, 
and  8  days,  at  7  per  cent.  1 

61.  What  is  the  interest  of  87c£.  15s.  4d.  for  2  years,  11 
months,  3  days,  at  1^  per  cent.  ? 

62.  What  is  the  interest  of  43=£.  16s.  for  9  months  and  13 
5,  at  8  per  cent.  ? 

63.  What  is  the  interest  of  142£.  19s.  for  1  year,  S 
months,  and  13  days,  at  9  per  cent.  ? 

64.  What  is  the  interest  of  8372  for  4  years,  8  months, 
and  17  days,  at  7^  per  cent.  \ 

65.  What  is  the  interest  of  1  dollar  t'or  15  days  at  7  per 
cent.  1 

66.  What  is  the  interest  of  8-25  for  13  days,  at  7|  per  cent.  1 

67.  What  is  the  interest  of  ^.375  for  19  days,  at  11  per 
cent.  ? 

68.  What  is  the  interest  of  $1147  for  8  hours,  at  6  per 
cent.  ? 

69.  What  is  the  interest  of  137^.  lis.  for  11  days  at  9 
per  cent.  ? 

70.  What  is  the  interest  of  15s.  for  3  months,  at  8  per 
cent.  1 

71.  What  is  the  interest  of  \Q£.  7s.  8d.  for  2  months,  at 
12  per  cent.  1 

72.  What  is  the  interest  of  4s.  3d.  for  17  }^ars,  3  months, 
and  7  days,  at  8  per  cept.  ? 

73.  A  man  gave  a  note  13th  Feb.  1817,  for  $753,  interest 
at  6  per  cent.,  and  paid  on  it  as  follows:  19th.  Aug.  1817. 
$45;  27th  June,  1818,  $143;  19th  Dec.  1818,  $25;  1  Itli 
May  1819,  $100 ;  and  14th  Sept.  1820,  he  paid  the  rest, 
principal  and  interest.     How  much  was  the  last  payment  ? 

74.  A  note  was  given  17th  July,  1814,  for  $1432,  interest 
at  6  per  cent.,  and  payments  were  made  as  follows  ;  15th 
Sept.  same  year,  $150;  2d  Jan.  1815,  $130;  16th.  Nov 
1815,  $23;  11th  April,  1817,  $237  ;  15th  Aug.  1818,  $47. 
How  much  was  due  on  the  note,  principal  and  interest,  5th 
Feb.  1819] 


ARITHMETIC, 


PART  IL 


NUMERATION. 

V 

I.  A  single  thing  of  any  kind  is  called  a  unit  or  unity. 

Particular  names  are  given  to  the  different  collections  of 
units. 

A  single  unit  is  called       ------       One. 

If  to  one  unit  we  join  one  unit  more,  the  collection  is  call- 
ed tioo ;  that  is,  one  added  to  one  is  called  two,  or  one  an4 
one  are       ----------       Two. 

One  added  to  two  is  called  three ;  two  and  one  are   Three. 

One  added  three  its  called  yt^z/r ;  three  and  one  are   Four, 

One  added  io  four  is  called  J^ye  ;  four  and  one  are     Five. 

One  added  to  Jive  is  called  six  ;   five  and  one  are         Siz. 

One  added  to  six  is  called  seven  ;  six  and  one  are     Seven. 

One  added  to  seven  is  called  eight ;  seven  and  one 
are       ___-_-.----     Eight. 

One  added  to  eight  is  called  nine  ;  eight  and  one  are  Nine. 

One  added  to  nine  is  called  ten  ;  nine  and  one  are      Ten. 

In  this  manner  we  might  continue  to  add  units,  and  to 
give  a  name  to  each  different  collection.  But  it  is  easy  to 
perceive  that,  if  it  were  continued  to  a  great  extent,  it  would 
be  absolutely  impossible  to  remember  the  different  names ; 
and  it  would  also  be  impossible  to  perform  operations  on 
large  numbers.  Besides,  we  must  necessarily  stop  some- 
where ;  and  at  whatever  number  we  stop,  it  would  still  be 
possible  to  add  more  ;  and  should  we  ever  have  occasion  to 
do  so,  we  should  be  obliged  to  invent  new  names  for  them, 
and  to  explain  them  to  others.  To  avoid  these  inconve- 
niences, a  method  has  been  contrived  to  express  all  the  num- 
bers that  are  necessary  to  be  used,  with  very  few  names. 


108  ARITHMETIC.  Part.  'Z, 

The  first  ten  numbers  have  each  a  distinct  name.  The 
collection  of  ten  simple  units  is  then  considered  a  unit :  it  is 
called  a  unit  of  the  second  order.  We  speak  of  the  collec- 
tions of  ten,  in  the  same  manner  that  we  speak  of  simple 
units ;  thus  we  say  one  ten,  two  tens,  three  tens,  fo4ir  tens, 
five  tens,  six  tens,  seven  tens,  eight  tens,  nine  tens.  These 
expressions  are  usually  contracted  ;  and  instead  of  them  we 
say  ten,  twenty,  thirty,  forty,  fifty,  sixty,  seventy,  eighty, 
ninety. 

The  numbers  between  the  tens  are  expressed  by  adding 
the  numbers  beJow  ten  to  the  tens.  One  added  to  ten  is 
called  ten  and  one  ;  two  added  to  ten  is  called  ten  and  two ; 
three  added  to  ten  is  called  ten  and  three,  &c.  These  are 
contracted  in  common  language  ;  instead  of  saying  ten  and 
three,  ten  and  four,  &c.,  we  say  thirteen,  fourteen,  fifteen, 
sixteen,  seventeen,  eighteen,  nineteen.  These  names  seem 
to  have  been  formed  from  three  and  ten,  four  and  ten,  &c. 
rather  than  from  ten  and  three,  ten  and  four,  &c.,  the  num- 
ber which  is  added  to  ten  being  expressed  first.  The  sig- 
nification, however,  is  the  same.  The  names  eleven  and 
twelve,  seem  not  to  have  been  derived  from  one  and  ten,  two 
and  ten ;  although  twelve  seems  to  bear  some  analogy  to 
two.  The  names  onetecn,  twoteen^  would  have  been  more 
expressive  ;  and  perhaps  all  the  numbers  from  ten  to  twenty 
would  be  better  expressed  by  saying  ten  one,  ten  two,  ten 
three,  &-c. 

The  numbers  between  twenty  and  thirty,  and  between 
thirty  and  forty,  &.c.  are  expressed  by  adding  the  numbers 
below  ten  to  these  numbers ;  thus  one  added  to  twenty  is 
called  twenty-one,  two  added  to  twenty  is  called  twenty-two, 
&.C. ;  one  added  to  thirty  is  called  thirty-one,  two  added  to 
thirty  is  called  thirty-two,  &c. ;  and  in  the  same  manner 
forty-one,  forty-two,  fifty-one,  fifty-two,  &.c.  All  the  num- 
bers are  expressed  in  this  way  as  far  as  ninety-nine,  that  is 
nine  tens  and  nine  units. 

If  one  be  added  to  ninety-nine,  we  have  ten  tens.  We 
then  put  the  ten  tens  together  as  we  did  the  ten  units,  and 
this  collection  we  call  a  unit  of  the  third  order,  and  give  it  a 
name.     It  is  called  one  hundred. 

We  say  one  hundred,  two  hundreds,  &c.  to  nine  hundreds, 
in  the  same  manner,  as  we  say  one,  two,  three,  &c. 

The  numbers  between  the  hundreds  are  expressed  by  adft- 
ing  tens  and  units.      With  units,  tens,  and  hundreds  we 


T.  NUMERATION.  109 

can  express  nine  hundreds,  nine  tens,  and  nine  units  ;  which 
is  called  nine  hundred  and  ninety-nine.  If  one  unit  be 
added  to  this  number,  we  have  a  collection  of  ten  hundreds ; 
this  is  also  made  a  unit,  which  is  called  a  unit  of  the  fourth 
order;  and  has  a  name.     The  name  is  thousand. 

This  principle  may  be  continued  to  any  extent.  Every 
collection  often  units  of  one  order  is  made  a  unit  of  a  higher 
order  ;  and  the  intermediate  numbers  are  expressed  by  the 
units  of  the  inferior  orders.  Hence  it  appears  that  a  very 
few  names  serve  to  express  all  the  different  numbers  which 
we  ever  have  occasion  to  use.  To  express  all  the  numbers 
from  one  to  nine  thousand,  nine  hundred,  and  ninety-nine, 
requires,  properly  speaking,  hut  twelve  different  names.  It 
will  be  shown  liereailei,  tiiut  t.ie<e  L.velve  names  express  the 
numbers  a  great  deal  farther. 

Various  methods  have  been  invented  for  writing  numbers, 
which  are  more  expeditious,  than  that  of  writing  their  names 
at  length,  and  which,  at  the  same  time,  facilitate  the  pro- 
cesses of  calculation.  Of  these  the  most  remarkable  is  the 
one  in  common  use,  in  which  the  numbers  are  expressed  by 
characters  called  ftgur(>.  This  method  is  so  perfect,  that 
no  better  can  be  expected  or  even  desired.  These  figures 
are  supposed  to  have  been  invented  by  the  Arabs  ;  hence 
they  are  sometimes  called  Arabic  figures.  The  figures  are 
nine  in  number.  They  are  exactly  accommodated  to  the 
manner  of  naming  numbers  explained  above.* 

♦  Next  to  the  Arabic  figures,  the  Roman  method  seems  to  be  the 
most  convenient  and  the  most  simple.  It  is  very  nearly  accommodat- 
ed to  the  mode  of  naming  numbers  explained  obove.  A  short  descrip- 
tion o^  it  may  be  interesting  to  some ;  and  it  will  often  be  found  ex- 
tremely useful  to  explain  this  method  to  the  pupil  before  the  other. 
The  pupil  will  understand  the  principles  of  this,  sooner  than  of  the 
other,  and  having  learned  this,  he  will  more  easily  comprehend  the 
other.  He  will  perfectly  comprehend  the  principle  of  carrying,  in  this, 
both  in  addition  and  subtraction,  and  the  similarity  of  this  to  the  com- 
mon method  is  so  striking  that  he  will  readily  understand  that  also. 

The  pupil  may  perform  some  of  the  examples  in  Sects.  I,  II,  and 
VIII,  Part  I,  with  Roman  characters. 

THE  ROMAN  NOTATION. 

One  was  written  with  a  single  mark,  thus,  I 

Two  was  written  with  two  marks         .         .  i. 

Three  was  written i'' 

Four  was  written         .....  IHI 
10 


no  ARITHMETIC.  Pari% 

One  is  written  -                        -             1 

Two  is  written  -          -              -      2 

Three  is  written  _           -             -             3 

Four  is  written  -          -               -      4 

Five  is  written  -           -            -             5 

Six  is  written  -          -              -      6 

'  Seven  is  written  -           -            -             7 

£J/^A^  IS  written  -          -              -      8 

iV/«e  is  written  -           -             -            9 

These  nine  figures  are  sometimes  called  the  9  digits.     By 

Five  was  written     .         .         .         .  .    HHI 

Six  was  written H"" 

Seven  was  written I!'llll 

Eiglit  was  written HIIIMI 

Nine  was  written 1 1  III  I II I 

Ten,  instead  of  being  written  with  ten  marks, 
was  expressed  by  two  marks  crossing  each 

other,  thus,  ' X 

vvhich  expressed  a  unit  of  the  second  order. 
Two  tens  or  twenty  were  written         .         .       XX 
Three  tens  or  thirty  were  written  .         .XXX 

And  so  on  to  ten  tens,  which  were  written  with  ten  crosses.  But  as  it 
was  found  inconvenient  to  express  numbers  so  large  as  seven  or  eight, 
with  marks  as  represented  above,  the  X  was  cut  in  two,  thus  X,  and 
the  upper  part  V  was  used  to  express  one  half  of  ten,  or  five,  and  the 
numbers  f.om  five  to  ten  were  expressed  by  writing  marks  after  the  V, 
to  express  tlie  number  of  units  added  to  five. 

Six  was  written       .         .         .         .         •  VI 

Seven  was  written       .         .         .         .  V|i 

Eigiit  was  written  .         .         .  V II I      . 

Nine  was  written Villi 

The  intermediate  numbers  between  the  tens  were  expressed  bj 
writing  the  excess  above  even  tens  after  the  tens. 

Eleven  was  written         .         .         .         .  XI 

Twelve  was  written  ....      Xll,&c. 

Twenty-seven  was  written  .         .  XXVI!,  &c. 

To  express  ten  Xs,  or  ten  tens,  that  is,  one  unit,  of  the  third  order, 
or  one  hundred,  three  marks  were  used,  thus,  C.  And  to  avoid  the  in- 
convenience of  writing  seven  or  eight  Xs,  the  C  was  divided,  thus  C, 
and  the  lower  part  L  used  to  express  five  Xs,  or  fifty. 

To  express  ten  hundreds,  four  dashes  were  used,  thus,  M.  Thislasi 
was  afterwards  written  in  this  form  CD  and  sometimes  CO,  and  waa 
then  divided,  and  13  was  used  to  express  five  hundreds. 

These  daslies  resemble  some  of  the  letters  af  the  alphabet,  and  those 
letters  were  afterwards  substituted  for  them. 

The  1  resembles  the  I ;  the  V  resembles  the  V  ;  the  X  resembles  the 
X  ,  thfi  L  resembles  the  L  ;  the  C  was  substituted  for  the  C  ;  the  13 
resembles  the  D  ;  and  the  M  resembles  the  M. 


NUMERATION. 


Ill 


t'nese  nine  characters  all  numbers  whatever  may  be  express- 
ed. 

To  express  ten,  we  make  use  of  the  first  character  1.  But 
to  distinguish  it  from  one  unit,  it  is  written  in  a  new  place, 
thus  10  ;  the  0,  which  is  called  zero  or  a  cipher,  being  plac- 
ed on  the  right.  The  zero  0  has  no  value,  it  is  used  only  to 
occupy  a  place,  when  there  is  nothing  else  to  be  put  in  that 
place. 

Numbers  expressed  with  the  Roman  Letters. 


One 

I 

Two 

II 

Three 

III 

Four     ' 

*IIII 

Five 

V 

Six 

VI 

Seven 

VII 

Eight 

VIII 

Nine 

*VIIII 

Ten 

X 

Eleven 

XI 

Twelve 

XII 

Thirteen 

XIII 

FourteeH 

*XIII1 

Fifteen 

XV 

Sixteen 

XVI 

Seventeen 

XVII 

Eighteen 

XVIIl 

Nineteen 

^XVIIII 

Twenty 

XX 

Twenty- one 

XXI 

Twenty-two 

XX  fl 

Twenty-three 

XXIII 

Twenty-four 

'^XXIIII 

Twenty-five 

XXV 

Twenty-six 

XXVI 

Twenty-seven 

XXVII 

Twenty- eiglit 

XXVIII 

Twen4;y-nine 

*XXVIIII 

Thirty 

XXX 

TJiirty-one 

XXXI 

Thirty-two 

XXXII,&c 

Forty 

*xxxx 

Fifty 

T 

Sixty 

LX 

Seventy 

LXX 

Eiglity 

LXXX 

Ninety 

"LXXXX 

One  hundred 

C 

Two  liundred 

cc 

Tliree  hundred 

ccc 

Four  hundred 

cccc 

Five  Jiundred 

D 

Six  iiuiuired 

DC 

Seven  hundred 

DCC 

Eight  iiundred 

DCCC 

Nino  hundred 

DCCCC 

One  thousand 

M 

One  thousand,  eight  hundred,  and  twenty-six     MDCCCXX  VI 

A  man  has  a  carriage  worth  seven  hundred  and  sixty-eigld  dollars  , 
and  two  horses,  one  worth  two  hundred  and  seventy-three  dollars,  and 
the  oth-er  worth  two  hundred  and  forty-seven  dollars;  how  many  dol- 
lars are  the  whole  worth  ? 

These  numoers  may  be  written  as  follows  : — 
Operation. 
DCCLXVIII  dolls.  ^       To  aiJd  these  numbers  together  it  is  easy 
CCLXXIII  dolls,  f  to  see  that  it  will  he  the  nio.-^t  convenient  to 
CCXXXXVII  dolls.   /  ccmmenceon  the  right,  and  coimt  tiie  Is 

V  first.     We  find   eiglit  of  them,  which  we 

MCCLXXXVllI  dolls.  J  should  write  thus  VIII,  but  observmg  that 

•  II  is  usual  to  write  four  IV,  instead  of  HIT,  and  nine  IX.  instead  of  Villi, 
and  forty  XL,  instead  of  XXXX,  ami  ninety  XC,  instead  of  I.XXXX,  «S:.c.  in 
whicl)  a  small  character  before  a  large,  lakes  out  its  value  from  llie  large. 
This  is  more  convenient  when  no  calculation  is  to  be  made.  But  when  they 
are  to  be  used  in  calcialalion,  the  melliod  given  in  the  text  is  best. 


112  i^RITHMETIC.  Part% 

Eleven  is  written  thus,  11,  with  two  Is.  The  1  on  the 
left  expresses  o?ie  #e« ;  and  the  one  on  the  right  expresses 
one  unit,  or  one  added  to  ten.  Twelve  is  written  12 ;  the 
1  on  the  left  signifies  one  ten,  and  the  2  on  the  right  sig- 
nifies two  units,  and  the  whole  is  properly  read  ten  and  two. 

there  are  more  Vs  we  set  down  only  III,  reserving  the  V  and  count- 
ing it  with  the  other  Vs.  Counting  tlie  Vs  we  find  two,  and  the  ontt 
which  we  reserved  makes  three.  Three  Vs  are  equivalent  to  one  X 
and  one  V.  We  write  the  V  and  reserve  the  X.  Counting  the  Xs, 
we  find  seven  of  them,  and  the  one  which  was  reserved  makes  eight. 
Eight  Xs  are  equivalent  to  LXXX.  We  write  the  three  Xs  and  re- 
serve the  L.  Counting  the  Ls,  we  find  two  of  them,  and  the  one 
which  was  reserved  makes  three.  Tliree  Ls  are  equivalent  to  CL. 
We  write  the  L  and  reserve  the  C,  Counting  the  Cs,  we  find  six  of 
them,  and  tiie  one  which  was  reserved  makes  seven.  Seven  Cs  are 
equivalent  to  DCC.  We  write  the  CC  and  reserve  the  D.  Count- 
ing the  Ds  we  find  one,  and  the  one  which  was  reserved  makes 
two.  Two  Ds  are  equivalent  to  M.  The  whole  sum  therefore  is 
MCCLXXXVIII  dollars. 

The  general  rule  for  addition,  therefore  is,  to  heginxcith  the  charac- 
iers  which  express  the  loicest  numbers  and  count  all  of  each  kind  to- 
gether  without  regard  to  their  value,  only  observing  that  Jive  Is  make 
one  V,  and  that  two  Vs  make  one  X,  and  that  Jive  Xs  make  one  L, 
^c,  and  setting  them  doion  accordingly. 

A  man  having  one  hundred  and  seventy-eight  dollars,  paid  awajf 
seventy-nine  dollars  for  a  horse;  how  many  had  he  left.'' 
Operation. 
CLXXVIII  dolls.  "^       To  perform  this  operation  we  begin  at  the 
LXXVIllI  dolls.  {  right  hand,  and  take  the  Is  from  the  Is,  the 

—  I  Vs  from  the  Vs,  &c.     But  a  difficulty  imrne- 

LXXXXVIIII  dolls.  ''  diately  occurs,  for  we  cannot  take  IIII  from 
III ;  it  is  necessary  therefore  to  take  the  IIII  from  VIII,  that  is,  from 
IIIIIIII,  which  leaves  IIII ;  these  we  set  down.  Since  we  have  used 
the  V  in  the  upper  line,  it  will  be  necessary  to  take  the  V  in  the  lower 
line  from  one  of  the  Xs,  that  is  from  VV.  V  from  VV,  leaves  V, 
which  we  set  down.  Having  used  one  of  the  Xs,  there  is  but  one 
left.  We  cannot  take  XX  from  X,  we  must  therefore  use  the  L, 
which  is  equivalent  to  five  Xs,  which,  added  to  the  one  X,  raake 
XXXXXX ;  from  these  we  take  XX  and  there  remain  XXXX,  which 
we  set  down.  Since  the  L  in  the  upper  line  is  already  used,  it  is 
necessary  to  take  the  L  in  the  lower  line  from  the  C  which  is  equiva- 
lent to  LL  ;  one  L  taken  from  these,  leaves  L,  which  we  set  down. 
The  whole  remainder  therefore  is  LXXXXVIIII  dolls. 

Hence  the  general  rule  for  taking  one  number  from  another,  ex- 
pressed by  the  Roman  characters,  is,  to  begin  with  the  characters  ex- 
pressing the  lowest  numbers,  and  take  those  of  the  same  kind  from 
each  other,  when  practicable,  but  if  any  of  the  numbers  to  be  subtract' 
ed  exceed  those  from  tchich  they  are  to  be  taken,  a  character  of  tk& 
next  highest  order  xnvst  be  taken,  and  reduced  to  the  order  requiredy 
and  joined  with  the  others  from  ichich  the  subtraction  is  to  he  viade^ 

This  process  is  called  subtraction 


I. 


NUMERATION. 


113 


The  following  is  the  manner  of  writing  the  numbers  from 
nine  to  ninety-nine,  inclusive. 

The  first  column  contains  the  figures,  the  second  shows 
the  proper  mode  of  expressing  them  ni  words  and  the  way 
in  which  they  are  always  to  be  understood,  and  the  tinrd 
contains  the  names  which  are  comnonly  applied.  The 
common  names  are  expressive  of  their  signification,  but  not 
so  much  so  as  those  in  the  second  column. 


Figures. 

Proper  mode  of  expressing 

Common  Karnes 

them  in  words. 

10. 

One  Ten  or  simply  Te7i. 

Ten. 

11. 

Ten  and  one. 

Eleven. 

12. 

Ten  and  two.   ' 

Twelve. 

13. 

Ten  and  three. 

Thirteen. 

14. 

Ten  and  four. 

Fourteen. 

15. 

Ten  and  five. 

Fifteen. 

16. 

Ten  and  six. 

Sixteen. 

17. 

Ten  and  seveni 

Seventeen. 

18. 

Ten  and  eight. 

Eighteen. 

19. 

Ten  and  nine. 

Nineteen. 

20. 

Two  tens. 

Twenty. 

21. 

Two  tens  and  one. 

Twenty-one. 

22. 

Two  tens  and  two. 

Twenty-two. 

23. 

Two  tens  and  three. 

Twenty-three. 

24. 

Two  tens  and  four. 

Twenty-four. 

25. 

Two  tens  and  five. 

Twenty-five. 

26. 

Two  tens  and  six. 

Twenty-six. 

27. 

Two  tens  and  seven. 

Tweniy-seven. 

28. 

Two  tens  and  eight. 

Twenty-eight. 

29. 

Two  tens  and  nine. 

Twenty-nine. 

30. 

Three  tens. 

Thirty. 

31. 

Three  tens  and  one. 

Thirty-one. 

32, 

&;c. 

Three  tens  and  two. 

Thirty-two. 

40. 

Four  n>ns. 

Forty. 

41, 

&LC. 

Four  tons  and  one. 

Forty-one. 

50. 

Five  tens. 

Fifty. 

51, 

&c. 

Five  tens  and  one. 

Fifty-one. 

60. 

Six  tens. 

Sixty. 

Gl, 

&C. 

Six  tens  and  one. 

Sixty-one. 

70. 

Seven  tens. 

Seventy. 

71. 

&c. 

Seven  tens  and  one. 

Seventy-one. 

80. 

Eight  tens. 

Eighty 

SI. 

&c. 

Eight  tens  and  one. 
10* 

Eighty-one. 

114  ARITHMETIC.  Part  2. 

Fi<rures  Proper  mode  of  expressing       Common  Names . 

them  in  words. 

90.  Nine  tens.  Ninety. 

91,  &c.  Nine  tens  and  one.  Ninety-one. 
99.                      Nine  tens  and  nine.  Ninety-nine. 

Nine  tens  and  nine  or  ninety-nine  is  the  largest  number 
that  can  be  expressed  by  two  figures.     If  one  be  added  to 
nine  tens  and  nine,  it   makes  ten  tens,  or  one  hundred.     To 
express  one  hundred  we  use  the  first  figure  again  ;  but  in 
order  to  show  that  it  has  a  new  value,  it  is  put  in  anothei 
place,  which  is  called  the  hundreds'  place.     The  hundreds' 
place  is  the  third  place  counting  from  the  right.     One  hun- 
dred is  written,  109  ;  two  hundred  is  written,  200  ;  three 
hundred  is  written,  300.     The  zeros  on  the  right  have  no 
value  ;  their  only  purpose  is  to  occupy  the  two  first  places, 
so  that  the  figures  1,  2,  3,  &LC.  may  stand  in  the  third  place. 
The  figures  in  the  second  place,  we  observe,  have  the 
same  value  whether  the  first  place  be  occupied  by  a  zero  or 
by  a  figure  :  for  example,  in  20  and  in  23  the  2  has  precise- 
ly the  same  value  ;  it  is  two  tens  or  twenty  in  both.     In  the 
first  there  is  nothing  added  to  the  twenty,  and  in  the  second 
three  is  added  to  it. 

It  is  the  same  with  figures  in  the  third  place.      They 
have  the  same  value,  whether  the  two  first  places  are  occu 
pied  by  zeros  or  figures.     In  400,  403,  420,  and  435,  the  4 
has  the  same  value  in  each,  that  is  four  hundred.     The  value 
of  every  figure,  therefore,  depends  upon  its  place  as  counted 
from  the  right  towards  the  left.     A  figure  standing  in  the 
first  place  signifies  so  many  units ;  the  same  figure  standing 
in  the   second  place  signifies  so  many  tens  ;  and  the  same 
figure  standing  in  the  third   place  signifies  so  many  hun- 
dreds.    For  example,  333,  the  three  on  the  right  signifies 
three  units,  the  three  in  the  second  place  signifies  three  tens 
or  thirty,  and  the  3  in  the  third  place  signi^.es  three  hun- 
dreds.  The  number  is  read  three  hundreds,  three  tens,  and 
three,  or  three  hundred   and  thirty-three.     We  have  seen 
that   all  the  numbers  from  ten  to  twenty,  from  twenty  to 
thirty,  &/C.  are  expressed  by  adding  units  to  the  tens  ;  m  the 
same  manner  all  the  numbers  from  one  hundred  to  two  hun- 
dred, from  two  hundred  to  three  hundred,  &-c.  are  expressed 
by  addmg  tens  and  units  to  the  hundreds. — For  example,  to 
express  five  hundred  and  eighty-two,  we  write  five  hundreds, 
eight  tens,  and  two  units  thus,  582. 


1.  NUMERATION.  115 

The  largest  number  that  can  be  expressed  by  three  figures 
is  999,  nine  hundreds,  nine  tens,  and  nine  units,  or  nine 
hundred  and  ninety-nine.  If  to  this  we  add  one  unit  more, 
we  have  a  collection  of  ten  hundreds,  which  is  called  one 
thousand.  To  express  this,  the  1  is  used  again  ;  but  to 
show  that  it  expresses  1  thousand  it  is  written  one  place  far- 
ther to  the  left,  that  is,  in  the  fourth  place,  thus  1000.  Two 
thousand  is  written  2000,  and  so  on,  to  nine  thousand, 
which  is  written  9000.  The  intermediate  numbers  are  ex- 
pressed by  adding  hundreds,  tens,  and  units  to  the  thou- 
sands. 

It  is  easy  to  see  that  this  manner  of  expressing  numbers 
may  be  continued  to  any  extent.  Every  time  a  figure  is  re- 
moved one  place  to  the  left  its  value  is  increased  ten-fold, 
and  since  nothing  limits  the  number  of  places  which  we  may 
use,  there  can  be  no  number  conceived,  however  large, 
which  cannot  be  expressed  with  these  nine  characters. 

We  sometimes  call  the  figures  in  the  first  place  or  right 
hand  place,  units  of  the  Jirst  order ;  those  in  the  second 
place,  or  the  collection  of  tens,  units  of  the  second  ordtr ; 
those  in  the  third  place,  or  the  collection  of  hundreds,  units 
of  the  third  order,  d^c. 

The  following  table  exhibits  the  first  nine  places  or  orders, 
with  their  names,  and  contains  a  few  examples  to  illustrate 
them. 


116 


ARITHMETIC. 


Part  % 


Seven  units  or  seven  ... 

Tliree  tens,  or  thirty       -         -         .         . 
Four  tens  and  six  units,  or  forty-six 

Eight  hundreds 

Seven  hundreds  and   three  units,  or  seven 

hundred  and  three         .... 
Five  liundreds  and  four  tens,  or  five  hundred 

and  forty         ..... 
Six  hundreds,  five  tens,  and  eight  units,  or 
six  1  undred  and  fifty-eight 

Six  thousands 

Six  thousands  and  five  units 

Six  thousands  and  four  tens,  or  six  thousand 

and  forty         ..... 
Sii  th^^usan'is  and  four  tens  and  five  units,  or 

six  thous/.nd  and  forty-five 
Six  thousryids  and  seven  hundreds 
Six  thousand,  seven  hundred,  and  five 
Six  thousand,  seven  hundred,  and  forty     - 
Six  thousand,  seven  hundred,  and  forty-five 
Four  tens  of  thousands,  or  forty  thousand 
Forty  thousand  and  three         ... 
Forty  thousand,  five  hundred  and  three 
Forty-seven    thousand,    five    hundred,    and 

eighty  three  .... 

Four  hundred  and  twenty-six  thousand,  eight 

hundred  and  fifty-three  ... 

Thrfs  hundred  and  twenty-eight   millions 
four  hundred  and  thirty-five  thousand,  six 
hundred  and  eighty-seven 
Three  hundred  millions 
Tvventy  millions        -  .  -  . 

Eight  millions 

Four  hundred  thousand 

Thirty  thousand 

Five  tho  isand 

Six  hundred 

Eighty 

Seven 


CO  00  ^  as  en  .b. 


4 
4   2 


7    0 


7 

0 

0    0   0 
0   5   0 


L  NUMERATION.  117 

In  looking  over  the  above  examples  it  will  be  observed, 
that  the  three  first  places  on  the  right  have  distinct  names, 
'  iz.  units,  tens,  hundreds  ;  and  that  the  three  next  places  are 
all  called  thousands,  the  first  being  called  simply  thousands ; 
the  second,  tens  of  thousands;  the  third,  hundreds  of  thou- 
sands. In  the  same  manner  there  are  three  places  appro- 
priated to  millions,  and  distinguished  in  the  same  way,  viz. 
millions,  tens  of  millions,  hundreds  of  millions.  The  same 
is  true  of  all  the  other  names,  three  places  being  appropriat- 
ed to  each  name.  From  this  circumstance  it  is  usual  to  di- 
vide the  figures  into  periods  of  three  figures  each.  This 
division  very  much  facilitates  the  reading  and  writing  of 
large  numbers.  Indeed  it  enables  us  to  read  a  number  con- 
sisting of  any  number  of  figures,  as  easily  as  we  can  read 
three  figures.     This  is  illustrated  in  the  following  example. 

^       i 

o  .2  "^ 

=3  r3  c  m  S  ed 

'■3  ^  O  5  o  2  „ 

•  S  a  S  1-2  — :  o  ."5 

C?  (^  H  W  S  H  & 


(0  m  (c  CO  ca  <»  w 

"XJ  "^3  'O  T3  'Q  "^  n3 

Qj  o)  a>  (P  o)  o  4) 

U,  )-  i->  }-,  ^  t^  i^ 

c  ^  .ti  c  G  .ti  c  c;  .  -  c  c:  .ti  s  c  .  ti  c  c  .ti  -  s  ."ri 
3aic3a;c3(ys33a;j;3a;c;-<ys3a)c 

3  8  5,6  7  9,2  5  8,6  7  3,4  6  2,9  2  7,6  4  8 

We  have  only  to  make  ourselves  familiar  with  reading 
and  writing  the  figures  of  one  period,  and  we  shall  then  be 
able  to  read  or  write  as  many  periods  as  we  please,  if  we 
know  the  names  of  the  periods. 

It  is  to  be  observed  that  the  unit  of  the  first  period  is  sim- 
ply one  ;  the  unit  of  the  second  period  is  a  collection  of  a 
thousand  simple  units  ;  the  unit  of  the  third  period  is  a  col- 
lection of  a  thousand  units  of  the  second  period,  or  a  mil 
lion  of  simple  units ;  and  so  on  as  we  proceed  towards  the 
left,  each  period  contains  a  thousand  units/of  the  period  next 
precedmg  it. 

The  figures  of  each  period  are  to  be  lead  in  precisely  the 
same  manner  as  the  figures  of  the  rigl.t  hand  period.  At 
the  end  of  each  period,  except  the  right  hand  period,  the 
name  of  the  period  is  to  be  pronounced.     The  right  hand 


il8  ARITHMETIC.  Part  2. 

period  is  always  understood  to  be  units  without  mention  be 
mg  made  of  the  name. 

In  the  above  exainple,  the  right  hand  period  is  read,  six 
hundred  and  forty  eight  (units  being  understood.)  The  second 
period  is  read  in  the  same  manner,  nine  hundred  and  twenty- 
seven, — but  here  we  must  mention  the  name  of  the  period  at 
the  end  ;  we  say,  therefore,  nine  hundred  and  twenty-seven 
thousand.  If  we  wouki  put  the  two  periods  together,  we  begin 
on  the  left  and  say,  nine  hundred  and  twenty-seven  thou- 
sand, six  hundred  and  forty-eight.  The  third  period  is  read 
four  hundred  and  sixty-two, — adding  the  name  of  the  period, 
it  becomes  four  hundred  and  sixty-two  millions:  and  the 
three  periods  are  read  together,  four  hundred  and  sixty-two 
millions,  nine  hundred  and  twenty-seven  thousand,  six  hun- 
dred and  forty-eight. 

Beginning  at  the  left  hand  of  the  above  example,  the  seve- 
ral periods  are  read  separately  as  follows — three  hundred 
and  eignty-five;  six  hundred  and  seventy-nine;  two  hun- 
dred and  fifty-eight ;  six  hundred  and  seventy-three  ;  four 
hundred  and  sixty-two ;  nine  hundred  and  twenty-seven  ; 
six  hundred  and  forty-eight.  Giving  each  period  its  name 
and  putting  all  together  as  one  number,  it  becomes  three 
hundred  and  eighty-five  quintiUions  ;  six  hundred  and  se^ 
venty-nine  quadrillions ;  two  hundred  and  fifty-eight  tril- 
lions ;  Six  hundred  and  seventy-three  billions  ;  four  hundred 
and  sixty-two  millions  ;  nine  hundred  and  twenty-seven 
thousand;  six  hundred  and  forty-eight. 

The  namos  of  the  periods  are  derived  from  the  Latin  nu- 
merals, by  giving  them  the  termination  ilUon  and  making 
some  other  alterations,  so  as  to  render  the  pronunciation 
easy.  After  quintiUions  come  sextillions,  septillions,  octil- 
lions, nonillions,  decillions,  iindecillions,  duodecilHons,  tfyc. 

A  number  dictated  or  enunciated,  is  written  by  beginning 
at  the  left  hand,  and  proceeding  towards  the  right,  care  be- 
ing taken  to  give  each  figuie  '*3  proper  place.  If  any  place 
is  omitted  in  the  enunciation,  the  place  must  be  supplied 
with  a  zero.  If,  for  example,  the  number  were  three  hun- 
dred and  twenty-seven  thousand,  and  fifty-three  ;  we  observe 
that  the  highest  period  mentioned  is  thousands,  which  is  the 
second  period,  and  that  there  are  hundreds  mentioned  in 
this  period,  (that  is,  hundreds  of  thousands,)  this  period  is 
therefore  filled,  and  the  number  will  consist  of  six  places. 
We  first  write  3  for  the  three  hundred  thousand,  then  2  im- 


II.  ADDITION.  119 

ineiliately  after  it  for  the  twenty  thousand,  then  7  for  the 
seven  thousand  ;  there  were  no  hundreds  mentioned  in  the 
enunciation,  we  must  pu^  a  zero  in  the  hundreds'  place,  then 
5  for  the  tens,  and  3  for  the  units,  and  tlie  number  will 
siand  thus,  327,053. 

Let  the  number  he  fvfty-tlirce  millions,  forty  thousand,  six 
liundied  and  eight.  Millions  is  the  third  period,  and  tens  of 
millions  is  the  highest  place  mentioned,  hence  there  will  be 
but  two  places  occupied  in  the  period  of  millions,  and  the 
whole  number  will  consist  of  eight  places.  We  first  write 
53  for  the  millions.  In  the  period  of  thousands  there  is  only 
one  place  mentioned,  that  is,  tens  of  thousands,  we  must  put 
a  zero  in  the  hundreds  of  thousands'  place,  then  4  for  the  forty 
thousand,  then  a  zero  again  in  the  thousands'  place  ;  in  the 
next  period  we  write  6  for  the  six  hundred,  there  being  no 
tens  in  the  example  we  put  a  zero  in  the  tens'  place,  and 
then  8  for  the  eight  units,  and  the  whole  number  will  stand 
thus,  53,040,608. 

Whole  periods  may  sometimes  be  left  out  in  the  enuncia- 
tion. W  hen  this  is  the  case,  the  places  must  be  supplied  by 
zeros.  Great  care  must  be  taken  in  writing  numbers-,  to 
use  precisely  the  right  number  of  places,  for  if  a  mistake  of  a 
single  place  be  m.ade,  all  the  figures  at  the  left  of  the  mis- 
take, will  be  increased  or  diminished  tenfold.* 


ADDITION. 

II.  We  have  seen  how  numbers  are  formed  by  the  suc- 
cessive addition  of  units.  It  often  happens  that  we  wish  to 
put  together  two  or  more  numbers,  and  ascertain  what  num- 
ber they  will  form. 

A  person  bought  an  orange  for  5  cents,  and  a  pear  for  3 
ctnts  ;  how  many  cents  did  he  pay  for  both  ? 

*  The  custom  of  using  nine  characters,  and  consequently  tlie  tenfold 
ratio  of  the  places,  is  entirely  arbitrary;  any  other  number  of  figures 
mi;U'ht  be  used  by  giving  tlie  places  a  ratio  corresponding  to  the  num- 
ber of  finfures,  if  we  had  only  the  seven  first  figures  f  )r  exrrmple,  the 
ratio  of  the  places  would  be  eight  fold,  and  we  should  write  numbers, 
in  every  other  respect,  as  we  do  now.  It  would  be  necessary  to  re- 
ject the  names  eight  and  nine,  and  use  the  name  of  ten  for  eight. 
Twenty  would  correspond  to  th<3  present  sixteen:  and  one  hundred, 
1()  the  present  sixty-four,  fcc.  The  following  is  an  example  of  the 
eight  fold  ratio,  with  the  numbers  of  the  ten  fold  ratio  corresponding  to 
theiu. 


120  ARITHMETIC.  Part  2. 

To  answer  this  question  it  is  necessary  to  put  tcgeiher  the 
numbers  5  and  3.  It  is  evident  that  the  first  time  a  child 
undertakes  to  do  this,  he  must  take  one  of  the  numbers,  as  5, 
and  join  the  other  to  it  a  single  unit  at  a  time,  thus  5  and  1 
are  6,  6  and  1  are  7,  7  and  1  are  8  ;  8  is  the  sum  of  5  and 
3.  A  child  is  obliged  to  go  through  the  process  of  adding 
by  units  every  time  he  has  occasion  to  put  two  numbers  to- 
gether, until  he  can  remember  the  results.  This  however 
he  soon  learns  to  do  if  he  has  frequent  occasion  to  put  num- 
bers together.  Then  he  will  say  directly  that  5  and  3  are  8, 
7  and  4  are  11,  &lc. 

Before  much  progress  can  be  made  in  arithmetic,  it  is 
necessary  to  remember  the  sums  of  all  the  numbers  from  one 
to  ten,  taken  two  by  two  in  every  possible  manner.  These 
are  all  that  are  absolutely  necessary  to  be  remembered.  For 
when  the  numbers  exceed  ten,  they  are  divided  into  two  or 
more  parts  and  expressed  by  two  or  more  figures,  neither  of 
which  can  exceed  nine.  This  will  be  illustra^.ed  by  the  ex- 
amples which  follow. 

A  man  hoiight  a  coat  for  twenty-four  dollars,  and  a  hat 
for  eight  dollars.     How  much  did  they  both  come  to  ? 

In  this  example  we  have  8  dolls,  to 
add  to  24  dolls.  Here  are  twenty  dolls, 
and  four  dolls,  and  eight  dolls.  Eight 
and  four  are  twelve,  which  are  to  be  join- 
Ten  fold  Eight  fold  Ten  fold 
to                  1  Fifteen               15    corresp.  to     13 

2  Sixteen  16        -        -        14 

3  Seventeen         17     -        -      -     15 

4  Twenty  20         -         -        IC 
.    5  Thirty  30     -        -      -     24 

6  Forty  40        .        -        32 

-  7  Fifty  50    -        -      .     40 
8  Sixty  60        -         -        43 

-  91  Seventy  70     -         -       -     56 
10  One  hundred    100,  &c.       -        64 

-  11  One  thousand   1000        -       -    512 
.        .       12 

In  the  same  manner  if  we  had  twelve  figures,  the  places  woa!d  have 
been  in  a  thirteen  fold  ratio. 

The  ten  fold  ratio  was  probably  su^^gested  by  counting  the  fingers. 
This  is  the  most  convenient  ratio.  If  the  ratio  were  less,  it  wonld  re- 
quire a  larger  number  of  places  to  express  large  numbers.  If  the  ratio 
were  larger,  it  would  not  require  so  many  places  indeed,  but  it  would 
not  be  so  easy  to  perform  the  operations  as  at  present  on  account  of 
the  numbers  in  each  place  being  so  large. 


Operation 

Coat  24  dolls. 

Hat 

8  dolls. 

Both  32  dolls. 

Eieht  fold 

One 

1  corresp 

Two 

2 

Three 

3 

J'our 

4 

Five 

5    - 

Six 

6        - 

Seven 

7    - 

Ten 

10 

Eleven 

11     - 

Twelve 

12        . 

Thrrteer 

13    - 

Fourteen  14 

II.  ADDITION.  121 

ed  to  twenty.  But  twelve  is  the  same  as  ten  and  two,  there- 
fore we  may  say  twenty  and  ten  are  thirty  and  two  are  thirty- 
two. 

A  man  bought  a  cow  for  27  dolls,  and  a  horse  for  68  dolls. 
How  much  did  he  give  for  both  ? 

Operation. 

Cow       27  dolls.         In  this  example  it  is  proposed  to  add 

Horse    68  dolls,     together  27  and  68.     Now  27  is  2  tens 

—  and  7  units ;  and  68  is  6  tens  and  8 

Both       95  dolls,     units.     6  tens  and  2  tens  are  8  tens  ; 

and  8  units  and  7  units  are  15,  which  is  1  ten  and  5  units ; 

this  joined  to  8  tens  makes  9  tens  and  5  units,  or  9.5. 

A  man  bought  ten  barrels  of  cider  for  35  dolls.,  and  1  bar- 
rels of  flour  for  42  dolls.,  a  hogshead  of  molasses  for  36 
dolls.,  a  chest  of  tea  for  87  dolls.,  and  3  hundred  weight  of 
sugar  for  24  dolls.      What  did  the  whole  amount  to  1 

Operation. 
Cider  35  doHs.  In  this  example  there  are  five  num- 
Flour  42  dolls,  bers  to  be  adde;]  logtiher.  We  ob- 
Molasses  36  dolls,  serve  that  each  of  these  numbers  con- 
Tea  87  dolls,  sists  of  two  figures.  It  will  be  most 
Sugar       24  JoUs.     convenient  to  add  together  either  all 

the  units,  or  all  the  tens  first,  and  then 

Amount  224  dolls,  the  other.  Let  us  begin  with  the 
tens.  3  tens  and  4  tens  are  seven  tens,  and  3  are  10  tens, 
and  8  are  18  tens,  and  2  are  20  tens,  or  200.  Then  adding 
the  units,  5  and  2  are  7,  and  6  are  13,  and  7  are  20,  and  4 
are  24,  that  is,  2  tons  and  4  units  ;  this  joined  to  200  makes 
224. 

It  would  be  still  more  convenient  lo  begin  with  the  units, 
in  the  following  manner  ;  5  and  2  are  7,  and  6  are  13,  and 
7  are  20,  and  4  are  24,  that  is  2  tens  and  4  units  ;  we  may 
now  set  down  the  4  units,  and  reserving  the  2  tens  add  them 
with  the  other  tens,  thus  :  2  tens  (which  we  reserved)  and  3 
tens  are  5  tens,  and  4  are  9  tens,  and  3  are  12  tens,  and  8 
are  20  tens,  and  2  are  22  tens,  which  written  with  the  4 
units  make  224  as  before. 

A  general  has  three  regiments  under  his  command ;  in  the 
first  there  are  478  men  ;  in  the  second  564  ;  and  in  the  third 
693.     How  many  men  are  there  in  the  whole  1 
11 


122 


ARITHMETIC. 


Part  2. 


First  reg. 
Second  reg 
Third  reg. 

In  all 


Operation. 


478  men  In  this  example,  each  of  the 
564  men  numbers  is  divided  in^o  three 
593  men     parts,  hundreds,  tens,  and  units. 

To  add  these  together  it  is  most 

1,635  men  convenient  to  begin  with  the 
units  as  follows  ;  8  and  4  are  12,  and  3  are  15,  that  is,  1 
ten  and  5  units.  We  write  down  the  5  units,  and  reserving 
the  1  ten,  add  it  with  the  tens.  1  ten  (which  we  reserved) 
and  7  tens  are  8  tens,  and  6  are  14  tens,  and  9  are  23  tens, 
that  is,  2  hundreds  and  3  tens.  We  write  down  the  3  tens, 
and  reserving  the  2  hundreds  add  them  with  the  hundreds. 
2  hundreds  (which  we  reserved)  and  4  hundreds  are  6  hun- 
dr»eds.  and  5  are  11  hundreds,  and  5  are  16  hundreds,  that  is, 
1  thousand  and  6  hundreds.  We  write  down  the  6  hundreds 
in  the  hundreds'  place,  and  the  1  thousand  in  the  thousands' 
place. 

The  reserving  of  the  tens,  hundreds,  ^bz.  and  adding  them 
with  the  other  tens,  hundreds,  &c.  is  called  carrying.  The 
principle  of  carrying  is  more  fully  illustrated  in  the  following 
example. 

A  merchant  had  all  his  money  in  hills  of  the  following 
description,  one-dollar  bills,  ten-dollar  bills,  hundred-dollar 
bills,  thousand-dollar  hills,  S^c.  each  kind  he  kept  in  a  sepa- 
rate box.  Another  merchant  presented  three  notes  for  pay- 
ment, one  2,673  dollars,  another  849  dollars,  and  another 
756  dollars.  How  much  teas  the  amount  of  all  the  notes ; 
and  hoio  many  bills  of  each  sort  did  he  pay,  supposing  he  paid 
it  loith  the  least  possible  number  of  bills  1 

Operation. 


The  first  note  would  require  2 
of  the  tho'isand-dollar  bills  ;   6  of 
of  the  hundred-dollar  bills  ;  7  ten- 
dollar  bills  ;  and  3  one-dollar  bills. 
4       2       7       8         The  second  note  would  require  8 
of  the  hundred-dollar  bills ;  4  ten-dollar  bills  ;  and  9  one- 
dollar  bills.     The  third  note  would  require  7  of  the  hundred- 
dol'ar  bills ;  5  ten-dollar  bills,    and  6  one-dollar  bills.    Count- 


rfj 

fO 

'O 

CO 

tn 

O 

c 

c 

O 

^ 

3 

O) 

C 

H 

133 

H 

O 

2 

6 

7 

3 

8 

4 

9 

7 

5 

6 

11.  ADDITION.  123 

ing"  the  one-dollar  bills,  we  find  18  of  them.  This  may  be 
paid  with  1  ten-dollar  bill  and  8  one-dollar  bills ;  putting  this 
1  ten-dollar  bill  with  the  other  ten-dollar  bills,  we  find  17  of 
them.  This  may  be  paid  with  1  hundred-dollar  bill,  and  7 
ten-dollar  bills  ;  i)utting  this  one-hundred  dollar  bill  with  the 
other  hundred-dollar  bills,  we  find  22  of  tliern  ;  this  may  be 
paid  with  2  of  the  thousand-dollar  bills,  and  2  of  the  hun- 
dred-dollar bills  ;  putting  the  2  thousand-dollar  bills  with  the 
other  thousand-dollar  bills,  we  find  4  of  them.  Hence  the 
three  notes  may  be  paid  with  4  of  the  thousand-dollar  bills, 

2  of  the  hundred-dollar  bills,  7  ten-dollar  bills,  and  8  one- 
dollar  bills,  and  the  amount  of  the  whole  is  4,278  dollars. 

Besides  the  figures,  there  are  other  signs  used  in  arithme- 
tic, which  stand  for  words  or  sentences  that  frequently  occur. 
These  signs  will  be  explained  when  there  is  occasion  to  use 
them. 

A  cross  -f-  one  mark  being  perpendicular,  the  other  hori- 
zontal, is  used  to  express,  that  one  number  is  to  be  added  to 
another.  Two  parallel  horizontal  lines  r=  are  used  to  ex- 
press equality  between  two  numbers.  This  sign  is  generally 
read  is  or  we  equal  to.     Example  5  -|-  3  zr:  8,  is  read  5  and 

3  are  8 ;  or  3  added  to  5  is  equal  to  8  ;  or  5  more  3  is  equal 
to  8  ;  or  more  frequently  5  plus  3  is  equal  to  8  ;  plus  being 
the  Latin  word  for  more.  These  four  expressions  signify 
precisely  the  same  thing. 

Any  number  consisting  of  several  figures  may  sometimes 
be  conveniently  expressed  in  parts  by  the  above  method. 
Example,  2358  =  2000  -|-  300  -f  50  -f  8  =  1000  +  1200 
-f  140  4-  18. 

A  man  owns  three  farms,  the  first  is  loorth  4,673  dollars  ; 
the  second,  5,764  dollars  ;  and  the  third,  9,287  dollars.  How 
many  dollars  are  they  all  toorth  1 

Perhaps  the  principle  of  carrying  may  be  illustrated  more 
plainly  by  separating  the  different  orders  of  units  from  each 
other. 


124  ARITHMETIC.  Pa-r  % 

Operation. 
4673  may  be  written  4000  -f- 
5764  -  -  5000  4- 
9287        -        -         9000  4- 


600  + 

70 -f 

3* 

700-}- 

60  + 

4 

200  4- 

80  + 

7 

14  18000  +  1500  +  210  +  14 
21. 

15  . .  Placing  the  results  under  each  other,  we 

18 . . .  have                                          18,000 

+ 1,500 

19,724  +    210 

+       14 


=r  19,724 

In  this  example  the  sum  of  the  units  is  14,  the  sum  of  the 
tens  is  21  tens  or  210,  the  sum  of  the  hundreds  is  15  hun- 
dreds or  1,500,  the  sum  of  the  thousands  is  18  thousands  or 
18,000  ;  these  numbers  being  put  together  make  19,724. 

If  we  take  this  example  and  perform  it  by  carrying  the 
tens,  the  same  result  will  be  obtained,  and  it  will  be  per- 
ceived that  the  only  difference  in  the  two  methods  is,  that  in 
this,  we  add  the  tens  in  their  proper  places  as  we  proceed, 
and  in  the  other,  we  put  it  off  until  we  have  added  each 
column,  and  then  add  them  in  precisely  the  same  places. 
Operation. 

4,673  Here  as  before  the  sum  of  the  units  is  14, 
+5,764  but  instead  of  writing  14  we  write  only  the  4, 
+9,287     and  reserving  the  1  ten,  we  say  1  (ten,  which 

we  reserved)  and  7  are  8,  and  6  are  14,  and 

=19,724  8  are  22  (tens)  or  2  hundreds  and  2  tens ; 
sotting  down  the  2  tens  and  reserving  the  hundreds,  we  say, 
2  (hundreds,  which  we  reserved)  and  6  are  8,  and  7  are  15, 
and  2  are  17  (hundreds)  or  1  thousand  and  7  hundreds  ; 
writing  down  the  7  hundreds,  and  reserving  the  1  thousand, 
we  say,  1  (thousand,  which  we  reserved)  and  4  are  5,  and  5 
are  10,  and  9  are  19  (thousands)  or  1  ten-thousand  and  9 
thousands  ;  we  write  the  9  in  its  proper  place,  and  since 
there  is  nothing  more  to  add  to  the  1  (ten  thousand)  we 
write  that  down  also,  in  its  proper  place.  The  answer  is 
19,724  dollars. 

*  It  will  be  well  for  the  learner  to  separate,  in  this  way,  several  of 
the  examples  in  Addition,  because  this  method  is  frequently  used  fof 
illustration  in  other  parts  uf  the  book. 


11.  ADDl'llON.  125 

We  may  now  observe  another  advantage  peculiar  to  this 
method  of  notation.  It  is,  that  all  large  numbers  are  divided 
into  parts,  m  order  to  express  them  by  the  different  orders  of 
units,  and  then  we  add  each  different  order  separately,  and 
without  regard  to  its  name,  observing  only  that  ten,  in  an 
inferior  order,  is  equal  to  one  in  the  next  superior  order.  By 
this  means  we  add  thousands,  millions,  or  any  of  the  higher  or- 
ders as  easily  as  we  add  units.  If  on  the  contrary  we  had  as 
many  names  and  characters,  as  there  arc  numbers  which  we 
have  occasion  to  use,  the  addition  of  large  numbers  would 
become  extremely  laboriouss  The  other  operations  are  as 
much  facilitated  as  Addition,  by  this  method  of  notation. 

In  the  above  examples  the  numbers  to  be  added  have  been 
written  under  each  other.  This  is  nat  absolutely  necessary ; 
we  may  add  them  standing  in  any  other  manner,  if  we  are 
careful  to  add  units  to  units,  tens  to  tens,  &c.,  but  it  is 
generally  most  convenient  to  write  them  under  each  other, 
and  we  shall  be  less  liable  to  make  mistakes. 

In  the  above  examples  we  commenced  adding  the  numbers 
at  the  top  of  each  line,  but  it  is  easy  to  see  that  it  will  make 
no  difference  whether  we  begin  at  the  top  or  bottom,  since 
the  result  will  be  the  same  in  either  case. 

Proof.  The  only  method  of  proving  addition,  which  can 
properly  be  called  a  proof,  is  by  subtraction.  This  will  be 
explained  in  its  proper  place. 

The  best  way  to  ascertain  whether  the  operation  has  been 
correctly  performed,  is  to  do  it  over  again.  But  if  we  add 
the  numbers  the  second  time  in  the  same  order  as  at  first,  if 
a  mistake  has  been  made,  we  are  very  liable  to  make  the 
tame  mistake  again.  To  prevent  this,  it  is  better  to  add 
them  in  a  reversed  order,  that  is,  if  they  were  added  down- 
wards the  first  time,  to  add  them  upwards  the  second  time, 
and  vice  versa* 

*  The  method  of  omitting  the  upper  line  the  second  time,  and  then 
adding  it  to  the  sum  of  the  rest  is  liable  to  the  same  objection,  as  that 
of  adding  the  numbers  twice  in  the  same  order,  for  it  is  in  fact  the 
same  thing.  If  this  method  were  to  be  used,  if  would  be  much  bet- 
ter to  omit  the  lower  lino  instead  of  the  upper  one  when  they  are 
added  upward  ;  and  the  upper  line  when  added  downward.  This 
would  change  the  order  in  which  the  numbers  are  put  together. 

The  danger  of  making  the  same  mistake  is  this  :  if  in  adding  up  ji 
row  of  figures  we  should  somewhere  happen  to  say  20  and  7  are  35, 
if  we  add  it  over  again  in  the  same  way,  we  are  very  liable  to  say  so 
again.  But  in  adding  it  in  another  order  it  would  be  a  very  .singuJar 
coincidence  if  a  mistake  of  exactly  the  same  number  were  made. 
11  » 


126  ARITHMETIC.  Part  % 

From  what  has  been  said  it  appears,  that  the  operation  of 
addition  may  be  reduced  to  the  following 

Rule.  Write  down  the  numbers  in  the  most  convenient 
manner,  which  is  generally  so  that  the  units  may  stand  under 
tmits,  tens  under  tens,  Sf'c.  First  add  together  all  the  units, 
and  if  they  do  not  exceed  nine,  write  the  result  in  the  units' 
place  ;  hut  if  they  amount  to  ten  or  more  t  \an  ten,  reserve  the 
ten  or  tens,  and  write  down  the  excess  above  even  tens,  in  the 
units^  place.  Then  add  the  tens,  and  add  with  them  the 
tens  which  were  reserved  from  the  preceding  column  ;  reserve 
the  tens  as  before,  and  set  doion  the  excess,  and  so  on,  till  all 
the  columns  arc  added. 


MULTIPLICATION. 

III.  Questions  often  occur  in  addition  in   which  a 

number  is  to  be  added  to  itself  several  times. 

How  much  will  4  gallons  of  molasses  come  to  at  34  cents 
a  gallon  7 

34  cents  This  example  may  be  performed  very 

34  cents  easily  by  the  common  method  of  addition. 
34  cents  But  it  is  easy  to  see  that  if  it  were  required 
34  cents      to  find  the  price  of  20,  30,  or  100  gal- 

Ions,  the  operation  would  become  laborious 

Ans.  136  cents      on  account  of  the  number  of  times  the 
number  34  must  be  written  down. 

I  find  in  adding  the  units  that  4  taken  4  times  amounts  to 
1 6,  I  write  the  6  and  reserve  the  ten  ;  3  taken  4  times 
amounts  to  12,  and  1  which  I  reserved  makes  13,  which  I 
write  down,  and  the  whole  number  is  136  cents. 

If  I  have  learned  that  4  times  4  are  16,  and  that  4  times 
3  are  12,  it  is  plain  that  I  need  not  write  the  number  34  but 
once,  and  then  I  may  say  that  4  times  4  are  16,  reserving  the 
ten  and  writing  the  6  units  as  in  addition.  Then  again  4 
times  3  (tens)  are  12  (tens)  and  1  (ten  which  I  reserved) 
are  13  (tens.) 

Addition  performed  in  this  manner  is  called  Multiplica 
tion.  In  this  example  34  is  the  number  to  be  multiplied  or 
repeated,  and  4  is  the  number  by  which  it  is  to  be  multi- 
plied ;  that  is,  it  expresses  the  number  of  times  34  is  to  be 
taken. 


III.  MULTIPLICATION.  127 

The  number  to  be  multiplied  is  called  the  multiplicand, 
and  the  number  which  shows  how  many  times  the  multipli- 
cand is  to  be  taken  is  called  the  multiplier.  The  answer  or 
result  is  called  the  product.  They  are  usually  written  in 
the  following  manner  : 

34  multiplicand 
4  multiplier 

136  product. 

Having  written  them  down,  say  4  times  4  are  16,  write 
the  6  and  reserve  the  ten,  then  4  times  3  are  12,  and  1 
(which  was  reserved)  are  13. 

In  order  to  perform  multiplication  readily,  it  is  necessary 
to  retain  in  n  smory  the  sum  of  each  of  the  nine  digits  re- 
peated from  one  to  nine  times  ;  that  is,  the  products  of  each 
of  the  nine  digits  by  themselves,  and  by  each  other.  These 
are  all  that  are  absolutely  necessary,  but  it  is  very  convenient 
to  remember  the  products  of  a  much  greater  number.  The 
annexed  table,  which  is  called  the  table  of  Pythagoras,  con- 
tains the  products  of  the  first  twenty  numbers*  by  the  first 
ten. 


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III.  MULTIPLICATION.  129 

To  form  this  table,  write  the  numbers  1,  2,  3,  4,  &c.  as 

far  as  you  wish  the  table  to  extend,  in  a  line  horizontally. 
This  is  the  first  or  apper  row.  To  form  the  second  row, 
add  these  numbers  to  iiemselves,  and  write  them  in  a  row 
directly  under  the  first.  Thus  1  and  1  are  2  ;  2  and  2  are 
4  ;  3  and  3  are  6  ;  4  and  4  are  8 ;  &c.  To  form  the  third 
row,  add  the  second  row  to  the  first,  thus  2  and  1  are  3 ;  4 
and  2  are  6  ;  G  and  3  are  9  ;  8  and  4  are  12  ;  &c.  This 
will  evidently  contain  the  first  row  three  times.  To  form 
the  fourth  row,  add  the  third  to  the  first,  and  so  on,  till  you 
have  formed  as  many  rows  as  you  wish  the  table  to  contain. 

When  the  formation  of  this  table  is  well  understood,  the 
mode  of  using  it  may  be  easily  conceived.  If  for  instance 
the  product  of  7  by  5,  that  is,  5  times  7  were  required,  look 
for  7  in  th-^  upper  row,  then  directly  under  it  in  the  fifth 
row,  you  find  35,  which  is  7  repeated  5  times.  In  the  same 
manner  any  other  product  may  be  found. 

If  you  seek  in  the  table  of  Pythagoras  for  the  product  of  5 
by  7,  or  7  times  5,  look  for  5  in  the  first  row,  and  directly 
under  it  in  the  seventh  row  you  will  find  35,  as  before.  It 
appears  therefore  that  5  times  7  is  the  same  as  7  times  5. 
In  the  same  manner  4  times  8  are  32,  and  8  times  4  are  32 ; 
3  times  9  are  27,  and  9  times  3  are  27.  In  fact  this  will  be 
found  to  be  true  with  respect  to  all  the  numbers  in  the  table. 
From  this  we  should  be  led  to  suppose,  that,  whatever  be  the 
two  numbers  which  are  to  be  multiplied  together,  the  product 
will  be  the  same,  whichsoever  of  them  be  made  the  multi- 
plier. 

The  few  products  contained  in  the  table  of  Pythagoras 
are  not  sufficient  to  warrant  this  conclusion.  For  analogical 
reasoning  is  not  allowed  in  mathematics,  except  to  discover 
the  probability  of  the  existence  of  facts.  But  the  facts  are 
not  to  be  admitted  as  truths  until  they  are  demonstrated.  I 
shall  therefore  give  a  demonstration  of  the  above  fact ;  which, 
besides  proving  the  fact,  will  be  a  good  illustration  of  the 
manner  in  which  the  product  of  two  numbers  is  formed. 

There  is  an  orchard,  in  which  there  are  4  rows  of  trees, 
and  there  are  7  trees  in  each  row. 

If  one  tree  be  taken  from  each 

row,  a  row  may  be  made  consisting 

of  four  trees ;  then  one  more  taken 

from  each  row  will  make   another 

row  of  four  trees ;  and  since  there  are  seven  trees  in  each 


130  ARITHMETIC.  Part%, 

row,  it  is  evident  that  in  tliis  way  seven  rows,  of  four  trees 
each,  may  be  made  of  them.  But  the  number  of  trees  re- 
mains the  same,  which  way  soever  they  are  counted. 

Now  whatever  be  the  number  of  trees  in  each  row,  if  they 
are  all  alike,  it  is  plain  that  as  many  rows,  of  four  each,  can 
be  made,  as  there  are  trees  in  a  row.  Or  whatever  be 
the  number  of  rows  of  seven  each,  it  is  evident  that  seven 
rows  can  be  made  of  them,  each  row  consisting  of  a  number 
equal  to  the  number  of  rows.  In  fine,  whatever  be  the  num- 
ber of  rows,  and  whatever  be  the  number  in  each  row,  it  is 
plain  that  by  taking  one  from  each  row  a  new  row  may  be 
made,  containing  a  number  of  trees  equal  to  the  number  of 
rows,  and  that  there  will  be  as  many  rows  of  the  latter  kind, 
as  there  were  trees  in  a  row  of  the  former  kind. 

The  same  thing  may  be  demonstrated  abstractly  as  fol- 
lows :  6  times  5  means  6  times  each  of  the  units  in  5  ;  bu|t 

6  times  1  is  6,  and  6  times  5  will  be  5  times  as  much,  that 
is,  5  times  6. 

Generally,  to  multiply  one  number  by  another,  is  to  repeat 
the  first  number  as  many  times  as  there  are  units  in  the 
second  number.  To  do  this,  each  unit  in  the  first  must  bo 
repeated  as  many  times  as  there  are  units  in  the  second. 
But  each  unit  of  the  first  repeated  so  many  times,  makes  a 
number  equal  to  the  second  ;  therefore  the  second  number 
will  be  repeated  as  many  times  as  there  are  units  in  the  first. 
Hence  the  product  of  two  numbers  will  always  be  the  same, 
whichsoever  be  made  multiplier. 

What  will  2o^  pounds  of  meat  cost,  at  7  cents  per  pound? 

This  question  will  show  the  use  of  the  above  proposition  ; 
for  254  pounds  will  cost  254  times  as  much  as  1  pound  ;  but 
I  pound  costs  7  cents,  therefore  it  will  cost  254  times  7. 
But  since  we  know  that  254  times  7  is  the  same  as  7  times 
254,  it  will  be  much  more  convenient  to  multiply  254  by  7. 
It  is  easy  to  show  here  that  the  result  must  be  the  same  ;  for 
254  pounds  at  1  cent  a  pound  would  come  to  254  cents  ;  at 

7  cents  a  pound  therefore  it  must  come  to  7  times  as  much. 

Operation. 

254  Here  say  7  times  4  are  28  ;  reserv- 

7  ing  the  2  (tens)  write  the  8  (units) ; 

then  7  times  5  (tens)  are  35  (tens)  and 

Ans.  1778  cents.  2  (tens)  which  were  reserved  are  37 

(tens) ;  write  the  7  (tens)  and  reserve  the  3  (hundreds) ; 


111.  ,  MULTIPLICATION.  131 

then  7  times  2  (hundreds)  are  14  (hundreds)  and  3  which 
were  reserved  are  17  (liundreds).  The  answer  is  1778 
cents  ;  and  since  100  cents  make  a  dollar,  we  may  say  17 
dollars  and  78  cents. 

The  process  of  multiplication,  by  a  single  figure,  may  be 
expressed  thus  :  Multiply  each  figure  of  the  multiplicand  by 
the  multiplier,  beginning  at  the  right  hand,  and  carry  as  in 
addition. 


IV.     Wliat  will  24  oxen  come  to,  at  47  dollars  apiece  ? 

It  does  not  appear  so  easy  to  multiply  by  24  as  by  a  nam 
ber  consisting  of  only  one  figure  ;  but  we  may  first  find  the 
price  of  6  oxen,  and  then  4  times  as  much  will  be  the  price 
of  24  oxen. 

Operation. 

47 

6 

282  dolls,  price  of  6  oxen. 
4 


1128  dolls,  price  of  24  oxen. 

Or  thus     47 
4 


188  dolls,  price  of  4  oxen. 
6 

1128  dolls,  price  of  24  oxen. 

A  number  which  is  a  product  of  two  or  more  numbers  is 
called  a  composite  or  compound  number.  The  numbers, 
which,  being  multiplied  together,  produce  the  number,  are 
CdWeA  factors  of  that  number.  4  is  a  composite  number,  its 
factors  are  2  and  2,  because  2  times  2  are  4.  6  is  also  a 
composite  number,  its  factors  are  2  and  3.  The  numbers  8, 
9,  10,  12,  14,  15,  &-C.  are  composite  numbers;  some  of 
them  have  only  two  factory,  and  some  have  several.  The 
sign  X ,  a  cross,  in  vvhich  neither  of  the  marks  is  either  hori- 
zontal or  perpendicular,  is  used  to  express  multiplication. 
Thus  3  X  2  =r  6,  signifies  2  times  3  are  equal  to  6.  2x3 
X  5  =:  30,  signifies  3  times  2  are  (5,  and  5  times  6  are  30. 


132  ARITHMETIC.  Part  2. 

Numbers  which  have  several  factors,  may  be  divided  into 
a  number  of  factors,  less  than  the  whole  number  of  factors, 
in  several  ways.  24,  for  example,  has  4  factors,  thus,  2  X 
2  X  2  X  3  zrz  24.  This  may  be  divided  into  2  factors  and 
into  3  factors  in  several  different  ways.  Thus  4  X  6  =  24 ; 
2X2x6z=24;  3x8  =  24;  2xl2  =  24;2xOX 
2  =  24. 

When  several  numbers  are  to  be  multiplied  together,  it 
will  make  no  difference  in  what  order  they  are  multiplied, 
the  result  will  always  be  the  same. 

Wliat  will  he  the  price  of  5  loads  of  cider,  each  load  con 
taining  7  barrels,  at  4  dollars  a  barrel  ? 

Now  5  loads  each  containing  7  barrels,  are  3.5  barrels. 
35  barrels  at  4  dollars  a  barrel,  amount  to  140  dollars.  Or 
we  may  say  one  load  comes  to  28  dollars,  and  5  loads  will 
come  to  140  dollars.  Or  lastly,  1  barrel  from  each  load  will 
come  to  20  dollars,  and  7  times  20  are  140. 

Thus  7     Or    7     Or    5 
5  4  4 

35  28         20 

4  5  7 

140         140        140 


What  is  the  price  of  23  loads  of  hay,  at  34  dolls,  a  had  t 

34 
2 

68  dolls,  price  of  2  loads. 
238  dolls,  price  of  7  loads. 


34 

7 


3 


714  dolls,  price  of  21  loads. 
|-  68  dolls,  price  of  2  loads. 

=  782  dolls,  price  of  23  loads. 


MULTIPLICATION.  133 


Multiply  328  hy  1 12. 
1]2  =  4X  7X4 

328 
4 

1312  product  oy  4 

7 

9184  product  by  28 
4 

36736  product  by  112 
It  is  easy  to  see  that  we  may  multiply  by  any  other  num- 
ber in  the  same  manner. 

This  operation  may  be  expressed  as  follows.  To  multiply 
by  a  composite  number  •  Find  tioo  or  more  numbers,  ivhich 
being  multiplied  together  icill  produce  the  multiplier  ;  multi- 
ply the  multiplicand  hy  one  of  these  numbers,  and  then  that 
product  by  another,  and  so  on,  until  you  have  midtiplied  by  all 
the  factors,  into  which  you  had  divided  the  multiplier,  and 
the  last  product  loill  be  the  product  required. 

If  the  multiplier  be  not  a  composite  number,  or  if  it  can- 
not be  divided  into  convenient  factors  :  Find  a  composite 
number  as  near  as  possible  to  the  rnultiplier,  hut  smaller,  and 
multiply  by  it  according  to  the  above  rule,  and  then  add  as 
many  times  the  multiplicand,  as  this  number  falls  short  of 
the  multiplier. 

V.  I  have  shown  how  to  multiply  any  number  by  a  sin- 
gle fiorure  ;  and  when  the  multiplier  consists  of  several 
figures,  how  to  decompose  it  into  such  numbers  as  shall  con- 
tain but  one  figure.  It  remains  to  show  how  to  multiply  by 
any  number  of  figures ;  for  the  above  processes  will  not 
alwr»ys  be  found  convenient. 

The  most  simple  numbers  consisting  of  more  than  one 
figure  are  10,  100,  1000,  &c.  It  will  be  very  easy  to  multi- 
ply by  these  numbers,  if  we  recollect  that  any  figure  written 
in  the  second  place  from  the  right  signifies  ten  times  as 
many  as  it  does  when  it  stands  alone,  and  in  the  third  place, 
one  hundred  times  as  many,  and  so  on.  If  a  zero  be  annex- 
ed at  the  right  of  a  figure  or  any  number  of  figures,  it  is 
evident  that  they  will  all  be  removed  one  place  towards  tlie 
left,  and  consequently  become  ten  times  as  great ;  if  t\yo 
zeros  be  annexed  they  will  be  removed  two  places,  and  will 
be  one  hundred  time^  as  great,  &c.  Hence,  to  multiply  by 
12 

I 


134  ARITHMETIC.  Part  2. 

any  n 
right 
cand. 


any  number  consisting  of  1,  with  any  number  of  zeros  at  the 
right  of  it  it  is  sufficient  to  annex  the  zeros  to  the  multipli- 


1  X  10  r=  10         1  X  100  z=  100 

2  X  10  =:  20         3  X  100  r=  300 

3  X  10  nz  30        5  X  100  =  500 

27  X       10  =        270 

42  X     100  =      4200 

368  X   1000  =z  368000 

VI.  When  the  multiplier  is  20,  30,  40,  200,  300,  2000, 
4000,  &,c.  These  are  composite  members,  of  which  10,  or 
100,  or  1000,  &LC.  is  one  of  the  factors.  Thus  20  z=  2  X 
10 ;  30  =  3  X  10  ;  300  =  3  X  100  ;  &c.  In  the  same 
manner  387000  =:  387  X  1000. 

How  much  wiU  30  hogsheads  of  wine  come  to^  at  87  dollar^ 
per  hogshead  ? 

Operation. 

87 

261  dolls,  price  of  3  hhds. 
10 

2610  dolls,  price  of  30  hhds. 

More  simply  thus      87 
30 

2610dolls.  price  of  30  hhds. 

It  appears  that  it  is  sufficient  in  this  example  to  multiply 
by  3  and  then  annex  a  zero  to  the  product.  If  the  number 
of  hogsheads  had  been  300,  or  3000,  two  or  three  zeros  must 
have  been  annexed.  It  is  plain  also  that,  if  there  are  zeros 
on  the  right  of  the  multiplicand,  they  may  be  omitted  until 
the  midtiplication  has  been  performed,  and  then  annexed  to 
the  product. 


VII.  MULTIPLICATION.  135 

VII.     A  man  bought  2G  pipes  of  wine,  at  143  dollars  a 
pipe ;  how  much  did  they  come  to  7 
26  :=z  20  4-  6.      The  operation  may  be  performed  thus  : 
143 
6 

858  dolls,  price  of  6  pipes 
143 
20 

2860  dolls,  price  of  20  pipes 
4-     858  dolls,  price  of  6  pipes 

=  3718  dolls,  price  of  26  pipes 

The  operation  may  be  performed  more  simply  thus ; 
143 
26 


+ 


2860  dolls,  price  of  20  pipes 
858  dolls,  price  of  6  pipes 

=  3718  dolls,  price  of  26  pipes 
Or  multiplying  first  by  6  : 
143 
26 

858  dolls,  price  of  6  pipes 
+  2860  dolls,  price  of  20  pipes 

=  3718  dolls,  price  of  26  pipes 

If  the  wages  of  1  man  he  438  dollars  for  1  year,  what  wiU 
he  the  wages  of  234  men,  at  the  same  rate  ? 

Operation, 
438 
234 


'  87600  dolls,  wages  of  200  men 
-f  13140    do.    wages  of  30  men 
-|-     1752    do.    wages  of  4  men 


=102492  dolls,  wages  of  234  men 


136  ARITHMETIC.  Part  2. 

Or  thus  438 

234 


1752  dolls,  wages  of  4  men 
-j-  13140    do.    wages  of  30  men 
4-  87600    do.    wages  of  200  men 


=:  102492  dolls,  wages  of  234  men 
When  we  multiply  by  the  30  and  the  200,  we  need  not 
aniiex  the  zeros  at  all,  if  we  are  careful,  when  multiplying 
by  the  tens,  to  set  the  first  figure  of  the  product  in  the  ten's 
place,  and  when  multiplying  by  hundreds,  to  set  the  first 
figure  in  the  hundred's  place,  &C. 

Operation. 
438 
234 


1752 
1314. 

876.. 


102,492 

If  we  compare  this  operation  with  the  last,  we  shall  find 
that  the  figures  stand  precisely  the  same  in  the  two. 

We  may  show  by  another  process  of  reasoning,  that  when 
we  multiply  units  by  tens,  the  first  figure  of  the  product 
should  stand  in  the  tens'  place,  &.c. ;  for  units  multiplied  by 
tens  ought  to  produce  tens,  and  units  multiplied  by  hundreds, 
ought  to  produce  hundreds,  in  the  same  manner  as  tens  mul- 
tiplied by  units  produce  tens. 

If  it  take  853  dollars  to  support  a  family  one  year,  how 
many  dollars  idHI  it  take  to  support  207  such  families  the 
same  time  ? 

Operation. 

853  In  this  example  I  multiply  first  by  the  7 

207  units,  and  write  the  result  in  its  proper  place  ; 

then  there  being  no  tens,  I  multiply  next  by 

597]  the  2  hundreds,  and  write  the  first  figure  of 

1706  this  product  under  the  hundreds  of  the  firs! 

product ;  and  then  add  the  results  in  the  ordej- 

176571  in  which  they  stand. 


Vlir.  SUBTRACTION.  137 

The  general  rule  therefore  for  multiplying  by  any  number  of 
figures  may  be  expressed  thus  :  Multiply  each  figure  of  the 
multiplicand  by  each  figure  of  the  multiplier  separately,  tak- 
ing care  ivhen  multiplying  by  units  to  make  the  fii  st  figure 
of  the  result  stand  in  the  vnits^  place  ;  and  when  multiplying 
by  tens,  to  make  the  first  figure  stand  in  the  tens''  place  ;  and 
when  multiplying  by  hundreds,  to  make  the  first  figure  stand 
in  the  hundreds''  place,  Sfc.  and  then  add  the  several  products 
together. 

Note.  It  is  generally  the  best  way  to  set  the  first  figure 
of  each  partial  product  directly  under  the  figure  by  which 
you  are  multiplying.  ^ 

Proof  The  proper  proof  of  multiplication  is  by  division, 
consequently  it  cannot  be  explained  here.  There  is  also  a 
method  of  proof  by  casting  out  the  nines,  as  it  is  called.  But 
the  nature  of  this  cannot  be  understood,  until  the  pupil  is 
acquainted  with  division.  It  will  be  explained  in  its  proper 
place.  The  instructer,  if  he  chooses,  may  explain  the  use 
of  it  here. 


SUBTRACTION. 

VIII.  A  inan  having  ten  dollars,  paid  away  three  of 
them ;  how  many  had  he  left  7 

We  have  seen  that  all  numbers  are  formed  by  the  suc- 
cessive addition  of  units,  and  that  they  may  also  be  formed 
by  adding  together  two  or  more  numbers  smaller  than  them- 
selves, but  all  together  containing  the  sa/ne  number  of  units 
as  the  number  to  be  formed.  The  nu/nber,  10  for  example, 
may  be  formed  by  adding  3  to  7,  7  -^  3  =r  10.  It  is  easy  to 
see  therefore  that  any  number  ma/  be  decomposed  into  two 
or  more  numbers,  which  taken  together,  shall  be  equal  to 
that  number.  Since  7  -f-  3  ii=  10,  it  is  evident  that  if  3  be 
taken  from  10,  there  will  re^nain  7. 

The  following  examples,  though  apparently  difterent,  all 
'equire  the  same  operation,  as  will  be  immediately  perceived. 

A  man  having  1 0  sheep  sold  3  of  them  ;  hoio  many  had  he 
left  ?  Thai  is,  if  3  be  taken  from  10,  lohat  number  icill  re- 
tnain  ? 

12* 


138  ARITHMETIC.  Part  % 

A  man  gave  3  dollars  to  one  son,  and  10  to  another ;  how 
much  more  did  he  give  to  the  one  than  to  the  other  7  That 
is,  how  much  greater  is  the  number  10  than  the  number  3  ? 

A  man  owing  10  dollars,  paid  3  dollars  at  one  timCj  and 
the  rest  at  another ;  how  much  did  he  pay  the  last  time  ? 
TJiat  is,  how  much  must  be  added  to  3  to  make  10  1 

From  Boston  to  DcdJiam  it  is  10  miles,  and  from  Boston 
to  Roxhury  it  is  only  3  miles ;  ichat  is  the  difference  in  the 
two  distances  from  Boston  7 

A  boy  divided  10  apples  between  two  other  boys ;  to  one 
he  gave  3,  how  many  did  he  give  to  the  other  7  That  is,  ij 
10  be  divided  into  two  parts  so  that  one  of  the  parts  may  be 
3,  what  will  the  other  part  be  7 

It  is  evident  that  the  above  five  questions  are  all  answered 
by  taking  3  from  10,  and  finding  the  difference.  This  ope- 
ration is  called  subtraction.  It  is  the  reverse  of  addition. 
Addition  puts  numbers  together,  subtraction  separates  a 
number  into  two  parts. 

A  man  paid  29  dollars  for  a  coat  and  7  dollars  for  a 
hat,  how  much  more  did  he  pay  for  his  coat  than  for  his  hat  f 

In  this  example  we  have  to  take  the  7  from  the  29 ;  wo 
know  from  addition,  that  7  and  2  are  9,  and  consequently 
that  22  and  7  are  29 ;  it  is  evident  therefore  that  if  7  be 
taken  from  29  the  remainder  will  be  22. 

A  man  hnught  an  ox  for  47  dollars  ;  to  pay  for  it  he  gave 
a  coiv  worth  23  dollars,  and  the  rest  in  money ;  how  much 
money  did  he  ^ay  7 

Operation. 

Ox  47  dollars.     Cow  23  dollars. 

It  will  be  best  to  perform  thi  example  by  parts.  It  is 
plain  that  we  must  takt  the  twenty  from  the  forty,  and  tlie 
three  from  the  seven  ;  thn  is,  the  tens  from-  the  tens,  and 
the  units  from  the  units.  I  take  twenty  from  forty,  and 
there  remains  twenty.  I  then  take  three  from  seven,  and 
there  remains  four,  and  the  whole  remainder  is  twenty-four. 
Ans.  24  dollars. 

It  is  generally  most  convenient  to  write  the  numbers  un- 
der each  other.  The  smaller  number  is  usually  written 
under  the  larger.  Since  units  are  to  be  taken  from  units 
and  tens  from  .ens,  it  will  be  best  to  write  units  under  units 


VIII.  SUBTRACTION.  139 

tens  under  tens,  &c.  as  in  addition.  It  is  also  most  con- 
venient, and,  in  fact,  frequently  necessary,  to  begin  with  t^he 
units  as  in  addition  and  multiplication. 

Operation. 
Ox     47  dollars.  I  say  first  3  from  7,  and  there  will 

Cow  23  dollars.  remain   4.      Then   2  (tens)  from  4 

—  (tens)  and  there  will  remain  2  (tens), 

24  difference,  and  the  whole  remainder  is  24. 

A  man  having  62  sheep  in  his  flock,  sold  17  of  them ;  how 
many  had  he  then  1 

Operation. 
He  had  62  sheep         In  this  example  a  difficulty  immedi- 
Sold       17  sheep     ately  presents  itself,  if  we  attempt  to 

perform  the  operation  as  before  ;  for 

Had  left  45  sheep  we  cannot  take  7  from  2.  We  can, 
however,  take  7  from  62,  and  there  remains  55  ;  and  10 
from  55,  and  there  remains  45,  which  is  the  answer. 

The  same  operation  may  be  performed  in  another  way, 
which  is  generally  more  convenient.  I  first  observe,  that  62 
is  the  same  as  50  and  12  ;  and  17  is  the  same  as  10  and  7. 
They  may  be  written  thus  : 

62  =  50  +  12         That  is,  I  take  one  ten  from  the  six 
17  =  10  -|-    7     tens,   and  write  it  with  the  two  units, 

But  the  17  I  separate  simply  into  units 

45  :=::  40  +  5  and  tens  as  they  stand.  Now  I  can  take 
7  from  12,  and  there  remains  5.  Then  10  from  50,  and  there 
remains  40,  and  these  put  together  make  45.* 

This  separation  may  be  made  in  the  mind  as  well  as  to 
write  it  down. 

Operation. 

62  Here  I  suppose  1  ten  taken  from  the  6  tens, 
17  and  written  with  the  2,  which  makes  12.  I  say 
—  7  from  12,  5  remains,  then  setting  down  the  5,  I 
45  say,  1  ten  from  5  tens,  or  simply  1  from  5,  and 
there  remains  4  (tens),  which  written  down  shows  the  re- 
mainder, 45. 

The  taking  of  the  ten  out  of  6  tens  and  joining  it  with 
the  2  units,  is  called  borrowing  ten. 

*  Lei  the  pupil  perform  a  large  numbe/  of  examples  by  separating 
them  in  this  way,  when  he  first  commences  subtraction. 


140  ARITHMETIC.  Part  % 

Sir  Isaac  Newton  was  horn  in  the  year  1642,  and  he  died 
in  1 727  ;  hoio  old  was  he  at  the  time  of  his  decease  1 

It  is  evident  that  the  difference  between  these  two  num- 
bers must  give  his  age. 

Operation. 
J600 -1-120  + 7=:  1727 
1600  -f-    40  +  2  —  1642 

Ans.  80  +  5  :=      85  years  old. 

In  this  example  I  take  2  from  7  and  there  remains  5, 
which  I  write  down.  But  since  I  cannot  take  4  (tens)  from 
2  (tens,)  I  borrow  1  (hundred)  or  10  tens  from  the  7  (hun- 
dreds,) which  joined  with  2  (tens)  makes  12  (tens,)  then  4 
(tens)  from  12  (tens)  there  remains  8  (tens,)  which  I  write 
down.  Then  6  (hundreds)  from  6  (hundreds)  there  re- 
mains nothing.  Also  1  (thousand)  from  1  (thousand)  no- 
thing remains.     The  answer  is  85  years. 

A  man  bought  a  quantity  of  jiour  for  1.5,265  dollars,  and 
sold  it  again  for  23,007  dollars,  hoiu  much  did  he  gain  by 
the  bargain  ? 

Operation. 
23,007         Here  I  take  5  from  7  and  there  remains 
15,265     2  ;  but  it  is  impossible  to  take  6  (tens)  from 

0,  and  it  does  not  immediately  appear  where 

2  I  shall  borrow  the  10  (tens,)  since  there  is 
nothing  in  the  hundreds'  place.  This  will  be  evident,  how 
ever,  if  I  decompose  the  numbers  into  parts. 

Operation, 
10,000  +  12,000  -f  900  -f  100  -I-  7  =  23,007 
10,000  4-    5,000  -f  200  -f    00  -}-  5  =  15,265 

'  7,000  +  700+    40-P"2z=    7,742 

The  23,000  is  equal  to  10,000  and  13,000  ;  this  last  is 
equal  to  12,000  and  1,000  ;  and  1,000  is  equal  to  900  and 
100.  Now  I  take  5  from  7,  and  there  remains  2  ;  60  from 
100,  or  6  tens  from  10  tens,  and  there  remains  40,  or  4 
tens;  2  hundreds  from  9  hundreds,  and  there  remains  7 
hundreds  ;  5  thousands  from  12  thousands,  and  there  re- 
mains 7  thousands  ;  and  1  ten-thousand  from  1  ten-thousand, 
and  nothing  remains.     The  answer  is  7,742  dollars. 

Tl;is  example  may  be  performed  in  the  same  manrfer  as 


VIII.  SUBTRACTION.  41 

the  others,  without  separating  it  into  parts  except  in  the 
mind. 

I  say  5  from  7,  there  remains  2 :  then  borrowing  10 
(which  must  in  fact  come  from  the  3  (thousand),  I  say,  6 
(tens)  from  10  (tens)  there  remains  4  (tens  ;)  tlien  I  borrow 
ten  again,  but  since  I  have  already  used  one  of  these,  I  say, 

2  (hundreds)  from  9  (hundreds)  there  remains  7  (hundreds  ;) 
then  I  borrow  ten  again,  and  having  borrowed  one  out  of  the 

3  (thousand,)  I  say,  5  (thousand)  from  12  (tliousand)  there 
remains  7  (thousand;)  then  1  (ten-thousand)  from  I  (ten- 
thousand)  nothing  remains.     The  answer  is  7,742  as  before. 

The  general  rule  for  subtraction  may  be  expressed 
thus :  The  less  number  is  always  to  be  subtracted  from  the 
larger.  Begin  at  the  right  hand  and  take  s-uccessivcli/  each 
figure  of  the  less  number  from  the  corresponding  figure  of 
the  larger  number^  that  is,  units  from  units,  tens  from  fens, 
^'c.  If  it  happens  that  any  figure  of  the  less  number  can- 
not be  taken  from  the  corre-^ponding  figure  of  the  larger^ 
borrow  ten  and  join  it  with  the  figure  from  which  the  subtrac- 
tion is  to  be  made,  and  then  subtract ;  before  the  next  figure 
is  subtracted  take  care  to  diminish  by  one  the  figure  from 
which  the  subtraction  is  to  be  made. 

N.  B.  When  two  or  more  zeros  intervene  in  the  number 
from  which  the  subtraction  is  to  be  made,  all,  except  the 
first,  must  be  called  9s  in  subtracting,  that  is,  after  having 
borrowed  ten,  it  must  be  diminished  by  one,  on  account  of 
the  ten  which  was  borrowed  before. 

Note.  It  is  usual  to  write  the  smaller  number  under  the 
greater,  so  that  units  may  stand  under  units,  and  tens  under 
tens,  &-C. 

Proof  A  man  bought  an  ox  and  a  cow  for  73  dollars^ 
and  the  price  of  the  cow  was  25  dollars  ;  what  was  the  price 
of  the  ox  1 

The  price  of  the  ox  is  evidently  what  remains  after  taking 
2.5  from  73. 

Operation. 
Ox  and  cow  73  dollars 
Cow  25     do. 

Ox  48     do. 

It  appears  that  the  ox  cost  48  dollars.  If  the  cow  cost  25 
dollars,  and  the  ox  48  dollars,  it  is  evident  that  25  and  48 
added  together  must  make  73  dollars,  what  they  botii  cost. 


142  ARITHMETIC.  Part  2. 

Hence  to  prove  subtraction,  add  the  remainder  and  the 
smaller  number  together,  and  if  the  work  is  right  their  sum 
will  be  equal  to  the  larger  number. 

Another  method.  If  the  ox  cost  48  dollars,  this  number 
taken  from  73,  the  price  of  both,  must  leave  the  price  of  the 
cow,  that  is,  2o.  Hence  subtract  the  remainder  from  the 
larger  number,  and  if  the*  work  is  right,  this  last  remainder 
will  be  equal  to  the  smaJler  number. 

Proof  of  addition.  It  is  evident  from  what  we  have  seen 
of  subtraction,  that  when  two  numbers  have  been  added  to- 
gether, if  one  of  these  numbers  be  subtracted  from  the  sum, 
the  remainder,  if  the  work  be  right,  must  be  equal  to  the 
other  number.  This  will  readily  be  seen  by  recurring  to  the 
last  example.  In  the  same  manner  if  more  than  two  num- 
bers have  been  added  together,  and  from  the  sum  all  the 
numbers  but  one,  be  subtracted,  the  remainder  must  be 
equal  to  that  one. 


DIVISION. 

IX.  A  hoy  having  32  apples  wished  to  divide  them  eqvxd- 
hj  among  8  of  his  companions ;  how  many  must  he  give  them 
apiece  ? 

If  the  boy  were  not  accustomed  to  calculating,  he  would 
probably  divide  them,  by  giving  one  to  each  of  the  boys,  and 
then  another,  and  so  on.  But  to  give  them  one  apiece  would 
take  8  apples,  and  one  apiece  again  would  take  8  more,  and 
so  on.  The  question  then  is,  to  see  how  many  times  8  may 
be  taken  from  32  ;  or,  which  is  the  same  thing,  to  see  how 
many  times  8  is  contained  in  32.  It  is  contained  four  times. 
Ans.  4  each. 

A  hoy  having  32  apples  was  able  to  give  8  to  each  of  his 
companions.     How  many  companions  had  he  7 

This  question,  though  different  from  the  other,  we  per- 
ceive, is  to  be  performed  exactly  like  it.  That  is,  it  is  the 
question  to  see  how  many  times  8  is  contained  in  32.  We 
take  away  8  for  one  boy,  and  then  8  for  another,  and  so  on. 

^  A  man  having  54  cents,  laid  them  all  out  for  oranges^  at 
6  cents  apiece.     How  many  did  he  huy  ? 


IX.  DIVISION.  143 

It  is  evident  that  as  many  times  as  6  cents  can  be  taken 
from  54  cents,  so  man}'  oranges  he  can  buy.  Ans.  9 
oranges 

A  man  bought  9  oranges  for  54  cents;  how  much  did  he 
give  apiece  1 

In  this  example  we  wish  to  divide  the  number  54  into  9 
equal  parts,  in  the  same  manner  as  in  the  first  question  we 
wished  to  divide  32  into  8  equal  parts.  Let  us  observe,  that 
if  the  oranges  had  been  only  one  cent  apiece,  nine  of  them 
would  come  to  9  cents  ;  if  they  had  been  2  cents  apiece, 
they  would  come  to  twice  nine  cents  ;  if  they  had  been  3 
cents  apiece,  they  would  come  to  3  times  9  cents,  and  so  on. 
Hence  the  question  is  to  see  how  many  times  9  is  contained 
in  54.     Ans.  6  cents  apiece. 

In  all  the  above  quesJons  the  purpose  was  to  see  how 
many  times  a  small  number  is  contained  in  a  larger  one,  and 
they  may  be  performed  by  subtraction.  If  we  examine  them 
again  we  shall  find  also,  that  the  question  was,  in  the  two 
first,  to  see  what  number  8  must  be  multiplied  by,  in  order 
to  produce  32  ;  and  in  the  third,  to  see  what  the  number  6 
must  be  multiplied  by,  to  produce  54  ;  in  the  fourth,  to  see 
what  number  9  must  be  multiplied  by,  or  rather  what  num- 
ber must  be  multiplied  by  9,  in  order  to  produce  54. 

The  operation  by  which  questions  of  this  kind  are  perform- 
ed is  called  division.  In  the  last  example,  54,  which  is  the 
number  to  be  divided,  is  called  the  dividend;  9,  which  is 
the  number  divided  by,  is  called  the  divisor ;  and  6,  which 
is  the  number  of  times  9  is  contained  in  54,  is  called  the 
quotient. 

It  is  easv  to  see  from  the  above  reasoning,  that  the  quo- 
tient and  divisor  multiplied  together  must  produce  the  divi- 
'dend  ;  for  the  question  is  to  see  how  many  times  the  divisor 
must  be  taken  to  make  the  dividend,  or  in  other  words  to  see 
what  the  divisor  must  be  multiplied  by  to  produce  the  divi- 
dend. It  is  evident  also,  that  if  the  dividend  be  divided 
by  the  quotient,  it  must  produce  the  divisor.  For  if  54  con- 
tains 6  nine  times,  it  will  contain  9  six  times. 

To  prove  division,  multiply  the  divisor  and  quotient  to- 
gether, and  if  the  work  be  right,  the  product  will  be  the 
dividend.  Or  divide  the  dividend  by  the  quotient,  and  if  the 
work  be  right,  the  result  will  be  the  divisor. 

This  also  furnishes'a  proof  for  multiplication,  for  if  the 


144  ARITHMETIC.  Part  2. 

quotient  multiplied  by  the  divisor  produces  the  dividend,  it 
is  evident,  that  if  the  product  of  two  numbers  be  divided  by 
one  of  those  numbers,  the  quotient  must  be  the  other  num 
ber. 

It  appears  that  division  is  applied  to  two  distinct  purposes, 
though  the  operation  is  the  same  for  both.  The  object  of 
the  first  and  fourth  of  the  above  examples  is  to  divide  the 
numbers  into  equal  parts,  and  of  the  second  and  third  to  find 
how  many  times  one  number  is  contained  in  another.  At 
present,  we  shall  confine  our  attention  to  examples  of  the 
latter  kind,  viz.  to  find  how  many  times  one  number  is  con- 
tained in  another. 

At  3  cents  apiece,  how  many  pears  may  he  bought  for  57 
cents  ? 

It  is  evident,  that  as  many  pears  may  be  bought,  as  there 
are  3  cents  in  57  cents.  But  the  solution  of  this  question 
does  not  appear  so  easy  as  the  last,  on  account  of  the  greater 
number  of  times  which  the  divisor  is  contained  in  the  divi- 
dend. If  w^e  separate  57  into  two  parts  it  will  appear  more 
easy. 

57  —  30  +  27. 

We  know  by  the  table  of  Pythagoras  that  3  is  contained 
in  30  ten  times,  and  in  27  nine  times,  consequently  it  is 
contained  in  57  nineteen  times,  and  the  answer  is  19  pears. 

How  many  harreh  of  cider,  at  3  dollars  a  barrel,  can  be 
bought  for  84  dollars  ? 

Operation. 
84  =z  60  -j-  24         3  is  contained  in  6  twice,  but  in  6 
tens  it  is  contained  ten  times  as  often,  or  20  times.     3  is 
contained  in  24  eight  times,  consequently  3  is  contained  28 
times  in  84.     Ans.  28  barrels. 

How  many  pence  are  there  in  1^2  farthings? 

As  many  times  as  4  farthings  are  contained  in  132  far- 
things, so  many  pence  there  are. 
Operation. 
132  =  120  -f-  12  120  is  12  tens,  4  is  contained  in 
12  three  times,  consequently  it  is  contained  30  times  in  12 
tens.  4  is  contained  3  times  in  12  units,  consequently  in 
132  it  is  contained  33  times.     Ans.  33  pence. 


IX.  DIVISION.  145 

How  many  barrels  of  four,  at  5  dollars  a  barrel,  may  be 
bought  for  785  dollars. 

Operation. 
7S5  =  500-1-  250  -f  35 

5  is  contained  in  5  once,  and  in  500  one  hundred  times. 
250  is  25  tens.  5  is  contained  5  times  in  25,  consequently 
50  times  in  259.  5  is  contained  7  times  in  35  units.  In 
785,  5  is  contained  157  times.     Ans.  157  barrels. 

How  many  dollars  are  there  in  7464  shillings  ? 

As  many  times  as  6  shillings  are  contained  in  7464  shil- 
lings, so  many  dollars  there  are. 
Operation. 
7464  =  6000  +  1200  -f-  240  +  24 

6  is  contained  1000  times  in  6000,  200  times  in  1200,  40 
limes  in  240,  and  4  times  in  24,  making  in  all  1244  times.* 
Ans.  1244  dollars. 

It  is  not  always  convenient  to  resolve  the  number  into 
parts  in  this  manner  at  first,  but  we  may  do  it  as  we  perform 
the  operation. 

In  126  days  lioio  many  weeks  7 
Operation. 
126  =  70  -f  56         Instead  of  resolving  it  in  this  man- 
ner, we  will  write  it  dowik  as  follows. 
Dividend  126     (7  Divisor 
70    — 

—  10 
56  8 
^Q    — 

—  18  quotient 

J.  ODserve  that  7  cannot  be  contained  100  times  in  126,  1 
therefore  call  the  two  first  figures  on  the  left  12  tens  or  120, 
rejecting  the  6  for  the  present.  7  is  contained  more  than 
once  and  not  so  much  as  twice  in  12,  consequently  in  12 
tens  it  is  contained  more  than  10  and  less  than  20  times.  I 
take  10  times  7  or  70  out  of  126,  and  there  remains  56. 
Then  7  is  contained  8  times  in  5Q,  and  18  times  in  126. 
Ans.  18  weeks. 

*  Let  the  pupil  perform  a  large  number  of  examples  in  this  manner 
when  he  first  commences  ;  as  he  is  obliged  to  separate  the  numbers 
into  parts,  he  will  at  length  come  to  the  common  method. 
13 


146  ARITHMETIC.  Part  2. 

Ifi  3756pencd  Jioio  ntamj  four-pences  1 
It  is  evident  that  this  answer  will  be  obtained  by  finding 
how  many  times  4  pence  is  contained  in  3756  pence. 

If  we  would  solve  this,  as  we  did  the  first  examples,  it 
will  stand  thus  : 

3756  —  3600  +  120  +  36 
But  if  we  resolve  it  into  parts,  as  we  perform  the  opera 
tion,  it  will  be  done  as  follows  : 
Dividend  3756  (4  divisor 

3600  

900  =  number  that  4  is  contained  in  3600 

156    30  do.       -        -        -        -        120 

120      9  do.  -        -        -        -      36 

36  939  do.       -        -        -        -      3756 

36 


Here  I  take  the  37  hundreds  alone,  and  see  how  many 
times  4  is  contained  in  it,  which  I  find  9  times,  and  since  it 
is  37  hundreds,  it  must  be  contained  900  times.  900  times 
4  is  3600,  which  I  subtract  from  3756,  and  there  remains 
156.  It  is  now  the  question  to  find  how  many  times  4  is 
contained  in  this.  I  take  the  15  tens,  rejecting  the  6,  and 
see  how  many  times  4  is  contained  in  it.  It  is  contained  % 
times,  and  since  it  is  15  tens,  this  must  be  3  tens  or  30 
times.  30  times  4  is  120.  This  I  subtract  from  156,  and 
there  remains  36.  4  is  contained  in  36,  9  times  ;  hence  it 
is  contained  in  the  whole  939  times.  Ans.  939  four-pences. 
If  these  partial  numbers,  viz.  3600,  120,  and  36,  are  com- 
pared with  the  resolution  of  the  number  above,  they  will  be 
found  to  be  the  same. 

This  operation  may  be  abridged  still  more. 
3756  (4 

36    

939  quotient 

15 
13 

36 
36 


IX.  DIVISION.  147 

In  this  I  say,  4  in»o  37  9  times,  and  set  down  the  9  in  the 
quotient,  without  regarding  whether  it  is  hundreds,  or  tens, 
or  units,  but  by  the  time  I  have  done  dividing,  if  I  set  the 
other  figures  by  the  side  of  it,  it  will  be  brought  into  its 
proper  place.  Then  I  say  9  times  4  are  36,  and  set  it  under 
the  37,  as  before,  but  do  not  write  the  zeros  by  the  side  of 
it.  I  then  subtract  36  from  37,  and  there  remains  1.  Tiiis 
of  course  is  100,  but  I  do  not  mind  it.  I  then  bring  down 
the  5  by  the  side  of  the  1,  which  makes  15,  or  rather  150, 
but  I  call  it  15.  Then  1  say,  4  into  15,  3  times,  (this  is  30, 
but  I  write  only  the  3;)  I  write  the  3  by  the  side  of  the  9. 
Then  I  say,  3  times  4  is  12,  which  I  write  under  the  15, 
and  subtract  it  from  15,  and  there  remains  3  (which  is 
in  fact  30.)  By  the  side  of  3  I  bring  down  the  6,  which 
makes  36.  Then  I  say  4  into  36,  9  times,  which  I  write 
in  the  quotient,  by  the  side  of  the  93,  and  it  makes  939. 
The  first  9  is  now  in  the  hundreds'  place,  and  the  3  in 
the  ten's  place,  as  they  ouglit  to  be.  If  this  operation  be 
compared  with  the  last,  it  will  be  found  in  substance  exactly 
iike  it.  AH  the  difference  is,  that  in  the  last  the  figures  are 
set  down  only  when  they  are  to  be  used. 

A  man  employed  a  number  of  workmen,  and  gave  them  27 
dollars  a  month  each ;  at  the  expiration  of  one  months  it  took 
10,125  dollars  to  pay  them.     Hoio  many  men  were  there  1 

It  is  evident  that  to  find  the  number  of  men  we  must  find 
how  many  times  27  dollars  is  contained  in  10,125  dollars. 

This  may  be  done  in  the  same  manner  as  we  did  the  last, 
though  it  is  attended  with  rather  more  difficulty,  because  the 
divisor  consists  of  two  figures. 

Operaiion. 

Dividend  10,125  (27  divisor 
8,100  

300  1=  the  number  of  times  27  is  contained 

2,025  in  8,100 

1,890     70  do.  -         -  1,890 

5  do.  -        -  135 

135  

135  375  do.        -        -        -     10.126 


14S  ARITHMETIC.  Part  2. 

Common  way, 

10,125  (27 

81       


375  quotient. 


202 
1S9 

135 
135 


I  observe  that  there  are  not  so  many  as  27  thousands,  so  I 
conclude  that  the  divisor  is  not  contained  1000  times  in  the 
dividend  ;  I  thv3refore  take  the  three  left  hand  figures,  neg- 
lecting the  other  two  for  the  present.  The  three  first  are 
101  ;  (properly  10,100,  but  I  notice  only  101  ;)  I  seek  how 
many  times  27  is  contained  in  101,  and  find  between  3  and 
4  times.  I  put  3  in  the  quotient,  which,  when  the  work  is 
done,  must  be  3  hundred,  because  101  is  101  hundreds,  but 
disregarding  this  circumstance,  I  find  how  much  3  times  27 
is,  and  write  it  under  101.  3  times  27  are  81  ;  this  subtract- 
ed from  101,  leaves  20.  By  the  side  of  20  I  bring  down 
2,  the  next  figure  of  the  dividend  which  was  not  used.  This 
makes  202,  for  the  next  partial  dividend.  I  seek  how  many 
times  27  is  contained  in  this.  I  find  7  times.  I  write  7  m 
the  quotient.  7  times  27  are  1S9,  which  I  subtract  from 
202,  and  find  a  remainder  13,  By  the  side  of  13  I  bring 
down  5,  the  other  figure  of  the  dividend,  which  makes  135 
for  the  last  partial  dividend.  I  find  27  is  contained  5  times 
in  this.  I  write  5  in  the  quotient.  5  times  27  are  135. 
There  is  no  remainder,  therefore  the  division  is  completed, 
Ans.  375  men. 

The  operation  in  the  above  example  is  precisely  the  same, 
as  in  those  which  precede  it ;  but  it  is  more  difficult  to  dis- 
cover how  many  times  the  divisor  is  contained  in  the  partial 
dividends.  When  the  divisor  is  still  larger,  the  difficulty  is 
increased.  I  shall  next  show  how  this  difficulty  may  be  ob- 
viated. 

In  31,755  days,  how  many  years,  alloiving  365  days  to  the 
year  ? 

It  is  evident,  that  as  many  times  as  365  is  contained  ip 
31,755,  so  many  years  there  will  be. 


IX.  DIVISION.  149 

Operation. 
Dividend  31755  {^^Gro  divisor 

2920      

87  quotient. 

2555 
2555 


I  observe  that  365  cannot  be  contained  in  317,  therefore  I 
must  take  the  four  left  hand  figures,  viz.  3175.  In  order  to 
discover  how  many  times  365  is  contained  in  this,  I  observe, 
that  365  is  more  than  300,  and  less  than  400.  I  say  300  is 
contained  in  3100,  or  simply  3  is  contained  in  31,  10  times, 
but  365  being  greater  than  300,  cannot  be  contained  in  it 
more  than  9  times.  Indeed  if  it  were  contained  more  than 
9  times,  it  must  have  been  contained  in  317,  which  is  impos- 
sible. 400  is  contained  in  3100,  (or  4  in  31)  7  times.  This 
is  the  iimit  the  other  way,  for  365  being  less  than  400,  must 
be  contained  at  least  as  many  times.  It  is  contained  there- 
fore 7,  or  8,  or  9  limes.  The  most  probable  are  8  and  9.  I 
try  9.  But  instead  of  multiplying  the  whole  number  365  by 
9,  I  say  9  times  300  are  2700,  or  simply  9  times  3  are  27  ; 
then  subtracting  2700  from  3170,  there  remains  470  ;  I  then 
say,  9  times  60  is  540,  or  simply  9  times  6  is  54,  which  being 
larger  than  470,  or  47,  shows  that  the  divisor  is  not  contain- 
ed 9  tim^s.  I  next  try  8  times,  and  say  as  before,  8  times 
300  are  2400,  which  subtracted  from  3170,  leaves  770,  then 
8  times  60  are  480,  which  not  being  so  large  as  770,  shows 
that  the  divisor  is  contained  8  times.  I  multiply  the  whole 
divisor  by  8  (which  is  in  fact  80,)  the  product  is  2920.  This 
subtracted  from  3175  leaves  255.  I  then  bring  down  the 
other  5,  which  makes  the  next  partial  dividend  2555.  Now 
trying  as  before,  I  fihd  that  3  is  contained  8  times  in  25, 
and  4  is  contained  6  times.  The  limits  are  6  and  8.  It  is 
probable  that  7  is  right.  I  multiply  365  by  7,  and  it  makes 
2555,  which  is  exactly  the  number  that  I  want.  If  I  had 
wished  to  try  8,  I  should  have  said  8  times  3  are  24,  which 
taken  from  25  leaves  1.  Then  supposing  1  to  be  placed 
before  the  next  figure,  which  is  5,  it  makes  15.  6  is  not 
contained  8  times  in  15,  therefore  365  cannot  be  contained 
8  times  in  2555.     The  answer  is  87  years. 

The  method  of  trying  the  first  figure  of  the  divisor  into 
the  first  ficrure,  or  the  first  two  figures  of  the  partial  dividend, 
13* 


150  ARITHMETIC.  Part  2. 

generally  enables  us  to  tell  what  the  quotient  figure  must 
be,  within  two  or  three,  and  it  will  always  furnish  the  limits. 
Then  if  we  try  the  second  figure,  we  shall  always  make  the 
limits  smaller  ;  if  any  doubt  then  remains,  which  will  not 
often  be  the  case,  we  may  try  the  tliird,  and  so  on. 

Divide  436940074  hy  64237. 

Operation. 
Dividend  436940074  (64237  divisor. 

385422*        

6802    quotient. 


.515180 
. 513896* 

. . . 128474 

.  . .  128474* 

Proof     436940074 

In  this  example  I  seek  how  many  times  6,  the  first  figure 
of  the  divisor,  is  contained  in  43,  the  first  two  figures  on  the 
left  of  the  dividend  ;  I  find  7  times,  and  7  is  contained  6 
times.  The  limits  are  6  and  7.  7  times  6  are  42,  and  42 
from  43  leaves  1,  which  I  suppose  placed  by  the  side  of  6  ; 
this  makes  16.  But  4,  the  second  figure  of  the  divisor,  is 
not  contained  7  times  in  16,  therefore  6  will  be  the  first 
figure  of  the  quotient. 

it  is  easy  to  see  that  this  must  be  6000,  when  the  division 
is  completed  ;  because  there  being  five  figures  in  the  divi- 
sor, and  the  first  figure  of  the  divisor  being  larger  than  the 
first  figure  of  the  dividend,  we  are  obliged  to  take  the  six 
first  figures  of  the  dividend  for  the  first  partial  dividend  ;  and 
the  dividend  containing  nine  figures,  the  right  hand  figure 
of  this  partial  dividend,  is  in  the  thousands'  place.  I  write 
6  in  the  quotient,  and  multiply  the  divisor  by  it,  and  write 
the  result  under  the  dividend,  so  that  the  first  figure  on  the 
right  hand  may  stand  under  the  sixth  figure  of  the  dividend, 
counted  from  the  left,  or  under  the  place  of  thousands.  This 
product,  subtracted  from  the  dividend  as  it  stands,  leaves  a 
remainder  51518  ;  by  the  side  of  this  I  bring  down  the  next 
figure  of  the  dividend,  which  is  0,  and  the  second  partial 
dividend  is  515180.  Trying  as  belore  with  the  6,  and  then 
with  the  4,  into  the  first  figures  of  this  partial  dividend,  I 
find  the  divisor  is  contained  in  it  8  (800)  times.     I  write  8 


IX.  DIVISION.  151 

in  the  quotient,  then  multiplying  and  subtracting  as  before,  I 
find  a  remainder  l"2S4.  I  bring  down  the  next  figure  of  the 
dividend,  which  gives  12847  for  the  next  partial  dividend.  I 
find  that  the  divisor  is  not  contained  in  this  at  all.  I  put  0 
in  the  quotient,  so  that  the  other  figures  may  stand  in  their 
proper  places,  when  the  division  is  completed.  Then  I  bring 
down  the  next  figure  of  the  dividend,  which  gives  for  a  par- 
tial dividend,  128474.  The  divisor  is  contained  twice  in 
this.  Multiplying  and  subtracting  as  before,  I  find  no  re- 
maindei.     The  division  therefore  is  completed. 

Proof.  It  was  observed  in  the  commencement  of  this 
Art.  that  division  is  proved  by  multiplying  the  divisor  by  the 
quotient.  This  is  always  done  during  the  operation.  In  the 
last  example,  the  divisor  was  first  multiplied  by  6  (0000,)  and 
then  by  8  (800,)  and  then  by  2  ;  we  have  only  to  add  these 
numbers  together  in  the  order  they  stand  in,  and  if  the  work 
is  right,  this  sum  will  be  the  dividend.  The  asterisms  show 
the  numbers  to  be  added. 

From  the  above  examples  we  derive  the  following  general 
rule  for  division  :  Place  the  divisor  at  the  right  of  the  divi- 
dend, separate  them  hy  a  mark,  and  draw  a  line  under  the 
divisor,  to  separate  it  from  the  quotient.  Take  as  many 
figures  on  the  left  of  the  dividend  as  are  necessary  to  contain 
the  divisor  once  or  more.  Seek  how  many  times  the  first  fig- 
ure of  the  divisor  is  contained  in  the  first,  or  two  first  figures 
of  these,  then  increasing  the  first  figure  of  the  divisor  hy  one, 
seek  how  many  times  that  is  contained  in  the  same  figure 
or  figures.  Take  the  figure  contained  within  these  limits, 
which  appears  the  most  probable,  and  multiply  the  two  left 
hand  figures  of  the  divisor  by  it ;  if  that  is  not  svjjicient  to 
determine,  multiply  the  third,  and  so  on.  When  the  first 
figure  of  the  quotient  is  discovered,  multiply  the  divisor  hy  it, 
and  subtract  the  product  from  the  partial  dividend.  Then 
write  the  next  figure  of  the  dividend  by  the  side  of  the  re- 
mainder. This  is  the  next  partial  dividend.  Seek  as  before 
how  many  times  the  divisor  is  contained  in  this,  and  place  the 
residt  in  the  quotient,  at  the  right  of  the  other  quotient  figure, 
then  multiply  and  subtract,  as  before ;  and  so  on,  until  all 
the  figures  of  the  dividend  have  been  used.  If  it  happens 
that  any  partial  dividend  is  not  so  large  as  the  divisor,  a  zero 
must  be  put  in  the  quotient,  and  the  next  figure  of  the  divi- 
dend icritten  at  the  right  of  the  partial  dividend. 

Note.     If  the  remainder  at  any  time  should  exceed  the 


152  ARITHMETIC.  Part  2. 

divisor,  the  quotient  figure  must  be  increased,  and  the  mul- 
tiplication and  subtraction  must  be  performed  again.  If  the 
product  of  the  divisor,  by  any  quotient  figure,  should  be 
larger  than  the  partial  dividend,  the  quotient  figure  must 
be  diminished. 


Short  Division. 

When  the  divisor  is  a  small  number,  the  operation  of  divi- 
sion may  be  much  abridged,  by  performing  the  multiplica- 
tion and  subtraction  in  the  mind  without  writing  the  results. 
In  this  case  it  is  usual  to  write  the  quotient  under  the  divi- 
dend.    This  method  is  called  skort  division. 

A  7nan  purchased  a  quantity  of  flour  for  3045  dollars,  at 
7  dollars  a  barrel.     How  many  barrels  were  there  ? 

Long  Division.  Short  Division. 

3045  (7  3045  (7 


28 


135  435 


24 
21 


35 
35 


In  short  division,  I  say  7  into  30,  4  times  ;  1  write  4  un- 
derneath ;  then  I  say  4  times  7  are  28,  which  taken  from  30 
leaves  2.  I  suppose  the  2  written  at  the  left  of  4,  which 
makes  24;  then  7  into  24.  3  times,  writing  3  underneath,  I 
say  3  times  7  are  21,  which  taken  from  24  leaves  3.  I  sup- 
pose the  3  written  at  the  left  of  5,  which  makes  35  ;  then  7 
in  35,  5  times  exactly  ;  I  write  5  underneath,  and  the  divi- 
sion is  completed. 

If  the  work  in  the  short  and  long  be  compared  together, 
they  will  be  found  to  be  exactly  alike,  except,  in  the  short  it 
is  not  written  down. 

X.  How  many  yards  of  cloth,  at  6  dollars  a  yard,  may 
be  bought  for  45  dollars  ? 


X.  DIVISION.  153 

42  dollars  will  buy  7  yards,  and  43  dollars  will  buy  8  yards. 
45  dollars  then  will  buy  more  than  7  yards  and  less  than 
8  yards,  that  is,  7  yards  and  a  part  of  another  yard.  As 
cases  like  this  may  frequently  occur,  it  is  necessary  to  know 
what  this  part  is,  and  how  to  distinguish  one  part  from 
another. 

When  any  thing,  or  any  number  is  divided  into  two  equal 
parts,  one  of  the  parts  is  called  the  /i«//of  the  thing  or  num- 
ber. When  the  thing  or  number  is  divided  into  three  equal 
parts,  one  of  the  parts  is  called  one  third  of  the  thing  or 
number  ;  when  it  is  divided  into  four  equal  parts,  the  parts 
are  called  fourths  ;  when  into  five  equal  parts,  ffths,  &lc. 
That  is  the  parts  always  take  their  names  from  the  number 
of  parts  into  which  the  thing  or  number  is  divided.  It  is 
evident  that  whatever  be  the  number  of  parts  into  which  the 
thing  or  number  is  divided,  it  will  take  all  the  parts  to  make 
the  whole  thing  or  number.  That  is,  it  will  take  two  halves, 
three  thirds,  four  fourths,  five  fifths,  &c.  to  make  a  whole 
one.  It  is  also  evident,  that  the  more  parts  a  thing  or  num- 
ber is  divided  into,  the  smaller  the  parts  will  be.  That  is, 
halves  are  larger  than  thirds,  thirds  are  larger  than  fourths, 
and  fourths  are  larger  than  fifths,  &c. 

When  a  thing  or  number  is  divided  into  parts,  any  num- 
ber of  the  parts  may  be  used.  When  a  thing  is  divided  into 
three  parts,  w^e  may  use  one  of  the  parts  or  two  of  them. 
When  it  is  divided  into  four  parts,  we  may  use  one,  two,  or 
three  of  them,  and  so  on.  Indeed  it  is  plain,  that,  when  any 
thing  is  divided  into  parts,  each  part  becomes  a  new  unit, 
and  that  we  may  number  these  parts  as  well  as  the  things 
themselves  before  they  were  divided. 

Hence  we  say  one  third,  two  thirds,  one  fourth,  two 
fourths,  three  fourths,  one  fifth,. two  fifths,  three  fifths,  &-c. 

These  parts  of  one  are  called  fractions,  or  broken  num- 
hers.  They  may  be  expressed  by  figures  as  well  as  whole 
numbers ;  but  it  requires  two  numbers  to  express  them,  one 
to  show  into  how  many  parts  the  thing  or  number  is  to  be 
divided  (that  is,  how  large  the  parts  are,  and  how  many  it 
takes  to  make  the  whole  one)  ;  and  the  other,  to  show  how 
many  of  these  parts  are  used.  It  is  evident  that  these  num- 
bers must  always  be  written  in  such  a  manner,  that  we  may 
know  what  each  of  them  is  intended  to  represent.  It  is 
agreed  to  write  the  numbers  one  above  the  other,  with  a 
jne  between  them.     The  number  below  the  line  shows  into 


154  ARITHMETIC.  Part  2. 

bow  many  parts  the  thing  or  number  is  divided,  and  the 
number  above  the  line  shows  how  many  of  the  parts  are 
used.  Thus  f  of  an  orange  signifies,  that  the  orange  is  divid- 
ed into  three  equal  parts,  and  that  two  of  the  parts  or  pieces 
are  used.  |  of  a  yard  of  cloth,  signifies  that  the  yard  is  sup- 
posed to  be  divided  into  five  equal  parts,  and  that  three  of 
these  parts  are  used.  The  number  below  the  line  is  called 
the  denominator,  because  it  gives  the  denomination  or  name 
to  the  fraction,  as  halves,  thirds,  fourths,  &c.  and  the  num- 
ber above  the  line  is  called  the  numerator,  because  it  shows 
how  many  parts  are  used. 

We  have  applied  this  division  to  a  single  thing,  but  it 
often  happens  that  we  have  a  number  of  things  which  we 
consider  as  a  bunch  or  collection,  and  of  which  we  wish  to 
take  parts,  as  we  do  of  a  single  thing.  In  fact  it  frequently 
happens  that  one  case  gives  rise  to  the  other,  so  that  both 
kinds  of  division  happ&n  in  the  same  question. 

If  a  barrel  of  cider  cost  2  dollars,  what  icill  ^  of  a  barrel 
cost  ? 

To  answer  this  question,  it  is  evident  the  number  two 
must  be  divided  into  two  equal  parts,  which  is  very  easily 
done.     I  of  2  is  1 . 

Again,  it  may  be  ashed,  if  a  barrel  of  cider  cost  2  dollars, 
what  part  of  a  barrel  loill  one  dollar  buy  1 

This  question  is  the  reverse  of  the  other.  But  we  have 
just  seen  that  1  is  i  of  2,  and  this  enables  us  to  answer  the 
question.     It  will  buy  j  of  a  barrel. 

If  a  yard  of  cloth  cost  3  dollars,  what  will  I  of  a  yard 
cost  1     What  ivill  ^  of  a  yard  cost  ? 

If  3  dollars  be  divided  into  3  equal  parts,  one  of  the  parts 
will  be  1,  and  two  of  the  parts  will  be  2.  Hence  \  of  a 
yard  will  cost  1  dollar,  and  f  will  cost  2  dollars. 

If  this  question  be  reversed,  and  it  be  asked,  what  part  of 
a  yard  can  be  bought  for  1  dollar,  and  what  part  for  2  dol- 
lars ;  the  answer  will  evidently  be  ^  of  a  yard  for  1  dollar, 
and  I  for  2  dollars. 

It  is  easy  to  see  that  any  number  may  be  divided  into  as 
many  parts  as  it  contains  units,  and  that  the  numoer  of  units 
used  will  be  so  many  of  the  parib  of  that  number.     Hence  if 


X.  DIVISION.  155 

it  be  asked,  what  part  of  5,  3  is,  we  say,  f  of  5,  because  I  i3 
}  of  5,  and  3  is  three  times  as  much. 

We  can  now  answer  the  question  proposed  above,  viz. 
How  many  yards  of  cloth,  at  6  dollars  a  yard,  may  be  bought 
for  45  dollars  ? 

42  dollars  will  buy  7  yards,  and  the  other  3  dollars  will 
buy  f  of  a  yard.  Ans.  7^  yards,  which  is  read  7  yards  and 
f  of  a  yard. 

A  man  hired  a  labourer  foi-  15  dollars  a  month  ;  at  the  end 
of  the  tune  agreed  upon,  he  paid  him  143  dollars.  How 
many  months  did  he  loork  7 

Operation. 
143  (15 
Price  of  9  months  135    — 

9y^5  months. 

Remainder  8 

The  wages  of  9  months  is  135  dollars,  which  subtracted 
from  143,  leaves  8  dollars.  Now  1  dollar  will  pay  for  ^j  of 
a  month,  consequently  8  dollars  will  pay  for  -j^  of  a  month. 
Ans.  9/3  months. 

Note.  The  number  which  remains  after  division,  as  8  in 
this  example,  is  called  the  remainder. 

At  97  dollars  a  ton,  how  many  tons  of  iron  may  he  bought 
for  2467  dollars  1 

Operation. 
2467  (97 

194    

25|f  tons. 

527 
485 

Remainder  42  dollars. 

After  paying  for  25  tons,  there  are  42  dollars  left.     1  dd- 
lar  will  buy  gV  o^a  ton,  and  42  dollars  will  buy  f|  of  a  ton. 
How  many  times  is  324  contained  in  18364  ? 
Operation. 
18364  (324 
1620    — — 

56||f  times    • 


2164 
1944 


Remainder  220 


156  ARITHMETIC.  Part  2. 

It  is  contained  56  times  and  220  over.  1  is  y|~j  of  324, 
and  220  is  ff f  of  324.  Ans.  56  times  and  |f f  of  another 
time. 

From  the  above  examples,  we  deduce  the  following  gene- 
lal  rule  for  the  remainder  :  When  the  division  is  performed, 
as  far  as  it  can  be,  if  there  is  a  remainder,  in  order  to  have 
the  true  quotient,  write  the  remainder  over  the  divisor  in  the 
form  of  a  fraction,  and  annex  it  to  the  quotient. 

XI.  We  observed  in  Art.  V.  that  when  the  multiplier  is 
10,  100,  1000,  &c.  the  multiplication  is  performed  by  an- 
nexing the  zeros  at  the  right  of  the  multiplicand.  In  like 
manner  when  the  divisor  is  10,  100,  1000,  &.c.  division  may 
be  performed  by  cutting  off  as  many  places  from  the  right  of 
the  dividend  as  there  are  zeros  in  the  divisor. 

At  \Q  cents  a  pound,  how  many  pounds  of  meat  may  he 
bought  for  64  cents  7 

The  6  which  stands  in  tens'  place  shows  how  many  times 
ten  is  contained  in  60,  for  60  signifies  6  tens,  and  the  4 
shows  how  many  the  number  is  more  than  6  tens,  therefore 
4  is  the  remainder.  The  operation  then  may  be  performed 
thus,  6.4.     The  answer  is  6y^  pounds. 

A  man  has  2347  lb.  of  tobacco,  which  he  wishes  to  put 
into  boxes  containing  100  lb.  each ;  how  many  boxes  will  it 
take  1 

It  is  evident  that  100  is  contained  in  2300,  23  times,  con- 
sequently it  will  take  23  boxes,  and  there  will  be  47  lbs. 
left,  which  will  fill  yVo  ^^  another  box.  The  operation  may 
be  performed  thus,  23.47.     Answer  23jyo. 

In  general  if  one  figure  be  cut  off  from  the  right,  the  tens 
will  be  brought  into  the  units'  place,  and  hundreds  into  the 
tens'  place,  &.c.  If  two  figures  be  cut  off,  hundreds  are 
brought  into  the  u»its'  place,  and  thousands  into  the  tens' 
place,  &c.  And  if  three  figures  be  cut  off,  thousands  are 
brought  into  the  units'  place,  &c.  that  is,  the  numbers  will 
be  made  10,  100,  or  1000  times  less  than  before. 

Hence  to  divide  by  10,  100,  1000,  S^c.  cut  off  from  the 
right  of  the  dividend  as  many  figures  as  there  are  zeros  in 
the  divisor.  The  remaining  figures  ivill  be  the  quotient,  and 
the  figures  cut  off  will  he  the  remainder,  ichich  ?nv<^t  be  writ' 
ten  over  the  divisor,  and  annexed  to  the  quotient. 


XII,  XIII.  DIVISION.  157 

XII.  We  observed  in  Art.  X,  that  any  two  numbers  be- 
ing given,  it  is  easy  to  teU  what  part  of  the  one  the  other  is. 
Thus  : 

What  part  of  10  i/ards  are  3  yards  1  Ans.  \  is  -^  cf  10, 
and  3  is  -^^  of  ten. 

What  part  of  237  barrels  is  82  barrels  ?  Ans.  1  is  ^\j 
of  237,  and  82  is  ^Vr  of  237. 

Fractions  are  properly  parts  of  a  unit,  but  by  extension 
the  term  fraction  is  often  applied  to  numbers  larger  than 
unity.  This  happens  when  the  numerator  is  larger  than  the 
denominator,  in  which  case  there  are  more  parts  taken  than 
are  sufficient  to  make  a  unit.  All  fractions  in  which  the 
numerator  is  equal  to  the  denominator,  as  |,  |,  i,  ^,  &c. 
are  equal  to  unity  ;  all  in  which  the  numerator  is  less  than 
the  denominator  are  less  than  unity,  and  are  called  proper 
fractions  ;  all  in  which  the  numerator  is  greater  than  the  de- 
nominator, are  more  than  unity,  and  are  called  improper 
fractions.     Thus  |,  y,  Y>  ^^^  improper  fractions. 

The  process  of  finding  what  part  of  one  number  another 
number  is,  is  called  finding  their  ratio. 

What  is  the  ratio  of  5  bushels  to  3  bushels,  or  of  5  to  3  1 
This  is  the  same  as  to  say,  what  part  of  5  is  3  ?  The  answer 
is  |.     The  ratio  of  5  to  3  is  |-. 

What  part  of  3  is  5  ?  Answer  4.  The  ratio  of  S  to  5 
is  f . 

What  is  the  ratio  of  35  yards  to  17  yards.     Answer  1^^. 

What  is  the  ratio  of  17  to  25  ?     Ansiocr  ^^. 

To  find  lohat  pan  of  one  number  another  is,  make  the 
number  lohich  is  called  the  part  (whether  it  be  the  larger  or 
smaller)  the  numerator  of  a  fraction^  and  the  other  number  the 
denominator. 

Also  to  find  the  ratio  of  one  number  to  another,  make  the 
number  lohich  is  expressed  first  the  denominator,  and  the 
other  the  numerator. 

XIII.  A  gentleman  gave  \  of  a  dollar  each  to  17  poor 
persons  ;  how  many  dollars  did  it  take  ? 

It  took  y  of  a  dollar.     But  f  of  a  dollar  make  a  dollar, 
consequently  as  many  times  as  5  is  contained  in  17,  so  many 
dollars  it  is.     5   is  contained  3  times  in  17,  aad  2  over 
14 


158  ARITHMETIC.  Part  2. 

That  is,   y  make  3  doliars,  and  there  are  f  of  another  dol- 
lar.    Ans.  3|  dollars. 

If  I  man  consume  -^^  of  a  barrel  of  flour  in  a  week,  how 
many  barr-els  will  an  army  of  537  men  consume  in  the  same 
time  1 

They  will  consume  M/.  |f  of  a  barrel  make  a  barrel, 
therefore  as  many  times  as  35  is  contained  in  537,  so  many 
barrels  it  is. 

537  (35 

35  

15i|-  barrel;.  Ans. 

175 

12 

35  is  contained  15  times  in  537  and  12  over,  which  is  ^| 
of  another  barrel. 

Numbers  like  3|,  15i|,  which  conta.in  a  whole  number 
and  a  fraction,  are  called  mixed  numbers.  The  above  pro- 
cess by  which  'y'  was  changed  to  3|-,  and  ^^-^  to  15^|^,  is 
called  reducing  improper  fractions  to  whole  or  mixed  num- 
bers. 

Since  the  denominator  always  shows  how  many  of  the 
parts  make  a  whole  one,  it  is  evident  that  any  improper  frac- 
tion may  be  reduced  to  a  whole  or  mixed  number,  by  the  fol- 
lowing rule  :  Divide  the  numerator  by  the  denominator,  and 
the  quotient  will  be  the  ichole  number.  If  there  be  a  remain- 
der, write  it  over  the  denominator,  and  annex  it  to  the  quo- 
tient, and  it  ivill  form  the  mixed  number  required. 

XIV.  It  is  sometimes  necessary  to  change  a  whole  or  a 
mixed  number  to  an  improper  fraction. 

A  man  distributed  3  dollars  among  some  beggars,  giving 
them  J  of  a  dollar  apiece ;  how  many  received,  the  money  7 
That  is,  in  3  dollars,  how  many  fifths  of  a  dollar  ? 

Each  dollar  was  divided  equally  among  5  persons,  conse- 
quently 3  dollars  were  given  to  15  persons.  That  is,  3  dol- 
lars are  equal  to  y  of  a  dollar. 

A  man  distributed  18|  bushels  of  wheat  among  some  poor 
persons,  giving  them  f  of  a  bushel  each ;  how  many  persons 
were  there  1 


XV.  DIVISION.  150 

This  question  is  the  same  as  the  following : 

In  18^  bushels,  hoio  many  if  of  a  bushel?  That  is,  how 
many  Iths  of  a  bushel  ?  ^ 

In  1  bushel  there  are  ^,  consequently  in  18  bushels  there 
are  18  times  7  sevenths  ;  that  is,  ^f-«,  and  j-  more  make  ^f «. 
Answer  129  persons. 

Reduce  28^  to  an  improper  fraction.  Tliaf  is,  in  28|^| 
hoto  many  ^'y. 

Since  there  are  ||  in  1,  in  28  there  must  be  28  times  as 
many.     28  times  25  are  700,  and  17  more  are  717.     Ans. 

717 

SI  ' 

Hence  to  reduce  a  whole  number  to  an  improper  fraction 
with  a  given  denominator,  or  a  mixed  number  to  an  improper 
fraction  :  multiply  the  whole  number  by  the  denominatnr^  and 
if  it  is  a  wAxed  number  add  the  numerator  of  the  fraction, 
and  write  the  result  over  the  denominator, 

XV.  A  man  hired  7  labourers  for  1  day  and  gave  them 
^  of  a  dollar  apiece ;  how  many  dollars  did  he  pay  the 
whole  1 

If  we  suppose  each  dollar  to  be  divided  into  5  equal  parts, 
it  would  take  3  parts  to  pay  I  man,  6  parts  to  pay  2  men, 
&c.  and  7  times  3  or  21  parts,  that  is,  ^j'  of  a  dollar  to  pay 
the  whole.     2_i  of  a  dollar  are  4}  dollars.     Ans.  4^  dollars. 

A  man  bought  13  bushels  of  grain,  at  ^  of  a  dollar  a 
bushel ;  how  many  dollars  did  it  come  to  ? 

I  of  a  dollar  are  5  shillings.  13  bushels  at  5  shillings  a 
bushel,  would  come  to  65  shillings,  which  is  10  dollars  and 
5  shillings. 

In  the  first  form,  13  times  I  of  a  dollar  are  y  of  a  dollar; 
that  is  lOf  dollars,  as  before. 

A  man  found  by  experience,  that  one  day  toith  another, 
his  horse  would  consume  ^  of  a  bushel  of  oats  in  a  day ;  how 
many  bushels  would  he,  consume  in  5  weeks  or  35  days  1 

If  we  suppose  each  bushel  to  be  divided  into  37  equal 
parts,  he  would  consume  13  parts  each  day.  In  35  days  he 
would  consume  35  times  13  parts,  which  is  455  parts.  But 
the  parts  are  37ths,  therefore  it  is  %y  =  12  \\  bushels. 


160 

ARITHMETIC. 

Part^ 

35 

13 

« 

105 
35 

This  process  is  called  multiplying  a  fraction  hy 

a  whole 

number. 

Multiply  tVtt  %  48. 

The  fraction  j^g  signifies  that  1  is  divided  into  1372 
equal  parts,  and  that  253  of  those  parts  are  used.  To  mul- 
tiply it  by  48,  is  to  take  48  times  as  many  parts,  that  is,  to 
multiply  the  numerator  253  by  48. 

253 


48 


2024 
1012 

12144 


¥tW  =  Si^fl 


The  product  of  253  by  48  is  12144  ;  this  written  over  the 
denominator  is  VVtV"'  which  being  reduced  is  8J-i^|.  Ans. 

To  multiply  a  fraction  then,  is  to  multiply  the  number  of 
parts  used  ;  hence  the  rule  :  multiply  the  numerator  and 
write  the  product  over  the  denominator. 

Note.  It  is  generally  most  convenient,  when  the  numera- 
tor becomes  larger  than  the  denominator,  to  reduce  the  frac- 
tion to  a  whole  or  mixed  number. 

It  is  sometimes  necessary  to  multiply  a  mixed  number. 

Bought  13  tons  of  iron,  at  97if  dollars  a  ton ;  what  did 
it  come  to  1 

In  this  example  the  whole  number  and  the  f/action  must 
be  multiplied  separately.  13  times  97  are  12G1.  13  times 
\^  are  ^V ,  equal  to  4ff ;  this  added  to  1261  makes  1265§4 
dollars.     Ans. 


XVI.  DIVISION.  ir»i 

Operation. 
97  14 

13  13 

^  42  Vt*  =  4H 

97  14 

1261  182 

1261  -f  4^A  —  126514  (lolls. 
Hence,  to  multiply  a  mixed  number  :  multiply  the  whole 
number  and  the  fraction  separately  ;  then  reduce  the  fraction 
to  a  tohole  or  mixed  number,  and  add  it  to  the  product  of  the 
whole  number. 

XVI.  We  have  seen  that  single  things  may  be  divided 
into  parts,  and  that  numbers  may  be  divided  into  as  many 
parts  as  they  contain  units  ;  that  is,  4  may  be  divided  into  4 
parts,  7  into  7  parts,  &.c.  It  now  remains  to  be  shown,  how 
every  number  may  be  divided  into  any  number  of  equal 
parts. 

If  3  yards  of  cloth  cost  12  dollars,  tohat  is  that  a  yard  ? 

It  is  evident  that  the  price  of  3  yards  must  be  divided  into 
3  equal  parts,  in  order  to  have  the  price  of  1  yard.  That  is, 
^  of  12  must  be  found. 

We  know  by  the  table  6f  Pythagoras,  that  3  times  4  are 
12,  therefore  |  of  12,  or  4  dollars  is  the  price  of  1  yard. 

If  5  yards  of  cloth  cost  45  dollars,  what  is  that  a  yard  ? 

1  yard  will  cost  \  of  45  dollars.  5  times  9  are  45,  there- 
fore 9  is  I  of  45,  or  the  price  of  1  yard. 

The  two  last  examples  are  similar  to  the  first  example 
Art.  9,  If  we  take  1  dollar  for  each  yard,  it  will  be  5  dol- 
lars, then  one  for  each  yard  again,  it  will  be  5  more,  and  so 
on,  until  the  whole  is  divided.  The  question,  therefore,  is 
to  see  how  many  times  5  is  contained  in  45,  and  the  result 
will  be  a  number  that  is  contained  5  times  in  45.  5  is  con- 
tained 9  times,  therefore  9  is  contained  5  times  in  45.  This 
IS  evident  also  from  Art.  III.  When  a  number,  therefore,  is 
to  be  divided  into  parts,  it  is  done  by  division.  The  number 
to  be  divided  is  the  dividend,  the  number  of  parts  the  divisor, 
and  the  quotient  is  one  of  the  parts. 
14* 


162  ARITHMETIC.  Part% 

A  man  owned  a  share  in  a  bank  worth  136  dollars,  and 
sold  ^  of  it ;  hoiv  many  dollars  did  he  sell  it  for  1 
136  (2 

Ans.  68  dollars. 

2  is  contained  68  times  in  136,  therefore  2  times  68  are 
136,  consequently  68  is  \  of  136. 

A  ticket  drew  a  prize  of  2,845  dollars,  of  which  A  ownnd 
\  ;  iDhat  was  his  share  ? 

2845  (5 

Ans.    569  dollars. 

Since  5  is  contained  569  times  in  2,845,  5  times  569  are 
equal  to  2,845,  therefore  569  is  |  of  2,845.  Division  may 
be  explained,  as  taking  a  part  of  a  number.  In  the  above  ex- 
ample 1  say,  i  of  28(00)  is  5(00)  and  3(00)  over.  Then 
supposing  3  at  the  left  of  4,  I  say,  }  of  34(0)  is  6(0)  and 
4(0)  over.  Then  ^  of  45  is  9.  Writing  all  together  it 
makes  569,  as  before.  The  same  explanation  will  apply 
when  the  divisor  is  a  large  number. 

Bought  43  tons  of  iron  for  4,171  dollars  ;  how  much  was 
it  a  ton  1 

1  ton  is  Jg  part  of  43  tons,  therefore  the  price  of  1  ton 
v/ill  be  -^  part  of  the  price  of  43  tons. 
4171  (43 


387 

301 
301 


97  dollars 


Two  men  A  and  B  traded  in  company  and  gained  456 
dollars,  of  which  A  was  to  have  f  and  B  f ;  what  was  the 
share  of  each  7 

The  name  of  the  fraction,  shows  how  to  perform  this  ex- 
ample. I  of  456  signifies  that  456  must  be  divided  into  8 
fHpial  parts,  and  5  of  the  parts  taken.  \  of  456  is  57,  5 
times  57  are  285,  and  3  times  57  are  171.  A's  share  285, 
and  B's  171  dollars. 


XVi.  DIVISION.  im 

456  (8  57 

3 

57  

5  B's  share  171  dollars. 

A's  share  285  dollars. 

A  man  bought  68  ban-els  of  pork  for  1224  dollars y  and 
sold  47  barrels,  at  the  same  rate  that  he  gave  for  it.  How 
much  did  the  47  barrels  come  to  1 

To  answer  this  question  it  is  necessary  to  find  the  price 
of  1  barrel,  and  then  of  47.  1  barrel  costs  ^  of  1224  dol- 
lars, and  47  barrels  cost  ^\  of  it.  ^V  of  1224  is  18.  47 
times  are  18  are  846.     Ans.  846  dollars. 

To  find  any  fractional  part  of  a  number,  divide  the  num- 
ber by  the  denominator  of  the  fraction^  and  multiply  the  quo- 
tient  by  ihe  numerator. 

A  man  bought  5  yards  of  cloth  for  28  dollars  ;  what  was 
that  a  yard  7 

\  of  25  is  5,  and  |  of  30  is  6.  \  of  28  then  must  be  be- 
tween 5  and  6. 

Cases  of  this  kind  frequently  occur,  in  which  a  number 
cannot  be  divided  into  exactly  the  number  of  parts  proposed, 
except  by  taking  fractions.  But  it  may  easily  be  done  by 
fractions. 

1  of  25  dollars  is  5  dollars.  It  now  remains  to  find  i  of 
3  dollars.  Suppose  each  dollar  divided  into  5  equal  parts, 
and  take  1  part  from  each.  That  will  be  3  parts  or  f  of  a 
dollar.  Ans.  5|  dollars.  |  of  a  dollar  is  ^  of  100  cents, 
which  is  60  cents.     Ans.  |5.60. 

A  man  had  853  lb,  of  butter,  which  he  ivished  to  divide 
into  7  equal  parts  ;  how  many  pounds  would  there  be  in  each 
part  ? 

-f  of  847  lb.  is  121  lb.  Then  suppose  each  of  the  6 
remaining  pounds  to  be  divided  into  7  equal  parts,  and  take 
1  part  from  each  ;  that  will  be  6  parts,  or  f  of  a  pound. 
Ans.  121f 

853  (7 

121f  lb.      Ans. 


164  ARITHMETIC.  Part  2. 

A  man  having  travelled  47  days^  found  that  he  had  travel- 
led 1800  miles ;  how  many  miles  had  he  travelled  in  a  day 
on  an  average  1  How  many  miles  would  he  travel  in  53 
days,  at  that  rate  ? 

In  one  day  he  travelled  ^'^  of  1800  miles,  and  in  53  days 
he  would  travel  f  |  of  it.  -^  of  1800  is  38,  and  14  over. 
^  of  1  is  Jy,  -^  of  14  is  14  times  as  much,  that  is,  ^|.  In 
one  day  he  travelled  38^  miles.  In  53  days  he  would 
travel  53  times  38^  miles. 

1800  (47  38  53 

141     53  14 

3S^  miles  in  1  day.  

390  114        212 

376  190  53 


14  2014        742 

Ans.      2029|^  miles  in  53  days. 

Hence  to  divide  a  number  into  parts  ;  divide  it  by  the 
number  of  parts  required^  and  if  there  be  a  remainder,  make 
it  the  numerator  of  a  fraction,  of  which  the  divisor  is  the  de- 
nominator. 

N.  B.  This  rule  is  substantially  the  same  as  the  rule  in 
Art.  X. 

When  one  part  is  found,  any  number  of  the  parts  may  be 
found  by  multiplication. 

It  was  shown  in  Art.  X.  that,  in  a  fraction,  the  denomina- 
tor shows  into  how  many  parts  1  is  supposed  to  be  divided, 
and  that  the  Lumerator  shows  how  many  of  the  parts  are  used. 
It  will  appear  from  the  following  examples,  that  the  numerator 
is  a  dividend,  and  the  denominator  a  divisor,  and  that  the 
fraction  expresses  a  quotient.  The  denominator  shows  into 
how  many  parts  the  numerator  is  to  be  divided.  In  this  man- 
ner division  may  be  expressed  without  being  actually  per- 
formed. If  the  fraction  be  multiplied  or  divided,  the  quo- 
tient will  also  be  multiplied  or  divided.  Hence  division  may 
be  first  expressed,  and  the  necessary  operations  performed  on 
the  quotient,  and  the  operation  of  division  itself  omitted,, 
until  the  last,  which  is  often  more  convenient.  Also,  wheu 
the  divisor  is  larger  than  the  dividend,  division  may  be  ex 
pressed,  though  it  cannot  be  performed. 


XVI.  DIVISION.  U>5 

A  gentleman  ivishes  to  divide  23  barrels  of  jlour  equaUy^ 
among  57  families  ;  huio  much  must  lie  give  them  apiece  ? 

In  this  example,  the  divisor  57  is  greater  than  the  divi- 
dend 23.  If  he  had  only  1  barrel  to  divide,  he  could  give 
them  only  j\  of  a  barrel  apiece  ;  but  since  he  had  23  bar- 
rels, he  can  give  each  23  times  as  much,  that  is,  -|^  of  a 
barrel. 

Hence  it  appears  that  §4  lightly  expresses  the  quotient  of 
23  by  57. 

If  it  be  asked  how  many  times  is  57  contained  in  23  1  It 
is  not  contained  one  time,  but  f|  of  one  time. 

If  \0  lbs.  of  copper  cost  3  dollars,  what  is  it  per  lb.  ? 

Here  3  must  be  divided  by  10.  J^  of  1  is  yV.  ^^'^  tV  ^^ 
3  must  be  f-^.     Ans.  ^^^^  of  a  dollar,  that  is,  30  cents. 

At  43  dollars  per  hhd.,  what  would  be  the  price  of  25  galls, 
of  gin  ? 

25  galls,  are  f  f  of  a  hogshead.  To  find  the  price  of  1 
gallon  is  to  find  -^  of  43  dolis.,  and  to  find  the  price  of  25 
galls,  is  to  find  ||  of  43  dolls.  ^  of  1  is  gL  ^  of  43  is  43 
limes  as  much,  that  is,  4|.  |^  is  25  times  as  much  as  j\f 
that  is,  25  times  ff.  25  times  f|  are  '^^  =  17g%  dolls. 
Ans. 

If  5  tons  of  hay  cost  138  dolls,  what  cost  3  tons  ? 

3  tons  will  C0st  |  of  138  dolls.  This  may  be  done  as  fol- 
lows. 1  of  138  is  27|,  and  3  times  27|,  are  82|  dolls. 
Ans.     Or, 

Expressing  the  division,  instead  of  performing  it,  ^  of  138 
is  i|«.  I  of  138  are  3  times  4^^  that  is,  ^-i-^  =:  82|  doIls» 
as  before. 

Note,  i  of  138  by  the  above  rule  is  27|.  But  the  same 
result  will  be  obtained,  if  we  say,  ^  of  138  is  i|^,  for  ^f® 
are  equal  to  27-^. 

The  process  in  this  Art.  is  called  multiplying  a  whole  num- 
ber by  a  fraction.  Multiplication  strictly  speaking  is  re- 
peating the  number  a  certain  number  of  times,  but  by  exten 
sion,  it  is  made  to  apply  to  this  operation.  The  definition 
of  multiplication,  in  its  most  extensive  sense,  is  to  take  one  ' 
number,  as  many  times  as  one  is  contained  in  another  mnn-- 
ber.  Therefore  if  the  multiplier  be  greater  than  1,  the  pro- 
duct will  be  greater  than  the  multiplicand  ;  but  if  the  multi- 


106  ARITHMETIC.  Part  2. 

pliei  be  only  a  part  of  1,  the  product  will  be  only  a  part  of 
the  multiplicand. 

It  was  observed  in  Art.  III.  that  when  two  whole  numbers 
are  to  be  multiplied  together,  either  of  them  may  be  made 
the  multiplier,  without  affecting  the  result.  In  the  same 
manner,  to  multiply  a  whole  number  by  a  fraction,  is  the 
same  as  to  multiply  a  fraction  by  a  whole  number. 

For  in  the  last  example  but  one,  in  which  43  was  multi- 
plied by  14,  25  and  43  were  multiplied  together,  and  the 
product  written  over  the  denominator  63,  thus  ^fy'.  The 
same  would  have  been  done,  if  ||  had  been  multiplied  by 
43. 

In  the  last  example  also,  138  was  multiplied  by  |.  The 
result  would  have  been  the  same  if  -2-  had  been  multiplied  by 
138. 

This  may  be  proved  directly. 

It  is  required  to  find  |f  of  43.  f|  of  1  is  f f ,  f^  of  43 
must  be  43  times  as  much,  that  is,  43  times  |f,  or  'fp  r= 
l7-g\.  So  also  I  of  1  is  I,  I  of  138  must  be  138  times  as 
much,  that  is,  138  times  |,  or  ^^  =  82|. 

Hence  to  multiply  a  fraction  hy  a  whole  number,  or  ai^hoh 
number  by  a  fraction ;  multiply  the  whole  number  and  the  nw- 
merator  of  the  fraction  together,  and  write  the  product  over 
the  denominator  of  the  fraction. 

XVII.  If  3  yards  of  cloth  cost  i  of  a  dollar,  what  is  that 
a  yard ? 

f  are  3  parts.    ^  of  3  parts  is  1  part.    Ans.  ^  of  a  dollar. 

A  man  divided  \^  of  a  barrel  of  flour  equally  among  4 
families  ;  how  much  did  he  give  them  apiece  ? 

If  are  12  parts.  J-  of  12  parts  is  3  parts.  Ans.  j^y  of  a 
barrel  each. 

This  process  is  dividing  a  fraction  by  a  whole  number.  A 
fraction  is  a  certain  number  of  parts.  It  is  evident  that  any 
number  of  these  parts  may  be  divided  into  parcels,  as  well  as 
the  same  number  of  whole  ones.  The  numerator  shows 
how  many  parts  are  used  ;  therefore  to  divide  a  fraction,  di- 
?yide  the  numerator. 

But  it  generally  happens  that  the  numerator  cannot  be 
exactly  divided  by  the  number,  as  in  the  folllowing  example. 

A  boy  icishes  to  divide  ^  of  an  orange  equally  between  two 
other  boys  ;  how  much  must  he  give  them  apiece  1 


XVII.  DIVISION.  167 

If  he  had  3  oranges  to  divide,  he  might  give  them  1  apiece, 
and  then  divide  the  other  into  two  equal  parts,  and  give  one 
part  to  each,  and  each  would  have  1^  orange.  Or  he  might 
cut  them  all  into  two  equal  parts  each,  which  would  make 
six  parts,  and  give  3  parts  to  each,  that  is,  -|  =  1^,  as  before. 
But  according  to  the  question,  he  has  |  or  3  pieces,  conse- 
quently he  may  give  1  piece  to  each,  and  then  cut  the  other 
into  two  equal  parts,  and  give  1  part  to  each,  then  each  will 
have  |-  and  i  of  i.  But  if  a  thing  be  cut  into  four  equal 
parts,  and  then  each  part  into  two  equal  parts,  the  whole  will 
be  cut  into  8  equal  parts  or  eighths ;  consequently  A  of  ^  is 
|-.  Each  will  have  i  and  ^  of  an  orange.  Or  he  may  cut 
each  of  the  three  parts  into  two  equal  parts,  and  give  ^  of 
each  part  to  each  boy,  then  each  will  have  3  parts,  that  is  |. 
Therefore  l  of  f  is  |.     Ans.  |. 

A  man  divided  ^  of  a  barrel  of  flour  equally  hettcecn  2  la- 
bourers ;  what  part  of  the  whole  barrel  did  he  give  to  each  1 

To  answer  this  question  it  is  necessary  to  find  ^  of  ]. 
If  the  whole  barrel  be  divided  first  into  5  equal  parts  or  fifths, 
and  then  each  of  these  parts  into  2  equal  parts,  the  whole 
will  be  divided  into  10  equal  parts.     Therefore,  \  of  \  is  y'^ 
He.  gave  them  -^^  of  a  barrel  apiece. 

A  man  owning  I  of  a  share  in  a  bank,  sold  i  of  his  part  ; 
what  part  of  the  whole  share  did  he  sell  1 

If  a  share  be  first  divided  into  8  equal  parts,  and  then  each 
part  into  3  equal  parts,  the  whole  share  will  be  divided  into 
24  equal  parts.  Therefore  |  of  |  is  '-^^,  and  i  of  |  is  7  times 
as  much,  that  is,  -^.     Ans.  /-p. 

Or  since  i  z=  ^,  |  =  |1,  and  \  of  |i  =  /^. 

In  the  three  last  examples  the  division  is  performed  by 
multiplying  the  denominator.  In  general,  if  the  denomina- 
tor of  a  fraction  be  multiplied  by  2,  the  unit  will  be  divided 
into  twice  as  many  parts,  consequently  the  parts  will  be  only 
one  half  as  large  as  before,  and  if  the  same  number  of  the 
small  parts  be  taken,  as  was  taken  of  the  large,  the  value  of 
the  fraction  will  be  one  half  as  much.  If  the  denominator 
be  multiplied  by  three,  each  part  will  be  divided  into  tiirce 
parts,  and  the  same  number  of  the  parts  being  taken,  the 
fraction  will  be  one  third  of  the  value  of  the  first.  Finally, 
if  the  denominator  be  multiplied  by  any  number,  the  parts 
will  be  so  many  times  smaller.     Therefore,  to  divide  afrac- 


168  ARITHxMETIC.  Part  2. 

tion,  if  the  numerator  cannot  be  dimded  exactly  by  the  divi- 
sor, multiply  the  denominator  by  the  divisor. 

A  man  divided  /g-  of  a  hogshead  of  wine  ijito  7  equal  parts, 
in  order  to  put  it  into  7  vessels  ;  what  part  of  the  whole  hogs- 
head did  each  vessel  contain  ? 

The  answer,  according  to  the  above  rule,  is  yf^.  The 
propriety  of  the  answer  may  be  seen  in  this  manner.  Siip- 
pose  each  16th  to  be  divided  into  7  equal  parts,  the  parts  will 
be  l]2ths.  From  each  of  the  /g-  take  one  of  the  parts,  and 
you  will  have  5  parts,  that  is  -r^-^' 

A  man  owned  -yj  of  a  ship^s  cargo ;  but  in  a  gale  the  cap- 
tain was  obliged  to  throjo  overboard  goods  to  the  amount  of 
^  of  the  whole  cargo.  What  part  of  the  loss  must  this  man 
sustain  7 

It  is  evident  that  he  must  lose  |  of  his  share,  that  is,  ^ 

-i  of  Jg  =:  y-J-2 ,  -g-  of  /j  :=  ^|^,  and  f  must  be  4  times  as 
much,  that  is,  -//2'     ^"s.  -f^,^  of  the  whole  loss. 

Or  it  may  be  said,  that  since  he  owned  -^-^  of  the  ship,  he 
must  sustain  -^  of  the  loss,  that  is,  j^  of  |^.  -^j  of -5  =  x|-2-, 
■jL.  of  I  =T'6"2'  ^"^  TT  ^s  7  times  as  much,  that  is,  ^^2*  ^^ 
before. 

This  process  is  multiplying  one  fraction  by  another,  and  is 
similar  to  multiplying  a  whole  number  by  a  fraction,  Art. 
XVI.  If  the  process  be  examined,  it  will  be  found  that  the 
denominators  were  multiplied  together  for  a  new  deiiomina- 
tor,  and  the  numerators  for  a  new  numerator.  In  fact  to  take 
a  fraction  of  any  number,  is  to  divide  the  number  by  the  de- 
nominator, and  to  multiply  tho  quotient  by  the  numerator. 
But  a  fraction  is  divided  by  multiplying  its  denominator,  and 
jnultiplied  by  multiplying  its  numerator.  We  have  seen  in 
the  above  example,  that  when  two  fractions  are  to  be  multi- 
plied, either  of  them  may  be  made  multiplier,  without  affect- 
ing the  result.  Therefore,  to  take  a  fraction  of  a  fraction, 
tluit  is,  to  multiply  one  fraction  by  another^  midtiply  tht  de- 
nenominators  together  for  a  new  denominator,  and  the  nume- 
rators for  a  new  numerator. 

If  7  dollars  icill  buy  5^  bushels  of  rye,  hoio  much  will  1 
dollar  buy  ?     Hotv  much  ivill  15  dollars  buy  ? 

1  dollar  will  buy  |  of  5-^  bushels.  In  order  to  find  ^  of  it, 
5f  must  be  changed  to  eighths.  5|  r=  \^,  I  of  V  =!?• 
I  dollar  will  buy  |f  of  a  bushel.     15  dollars  will  buy  15 


XVII.  DIVISION.  ir>() 

tiroes  as  much.     15  times  ^l  — ^.\^  ^IVJ.     Ans.   11^| 
bushels. 

If  13  bbls.  of  berf  cost  95^  dollars,  what  icill  25  bbh. 
cost  ? 

1  bbl.  will  cost  -Jj  of  95|-  dollars,  and  25  bbls.  will  cost-^-^ 
of  it.  To  find  this,  it  is  best  to  multiply  first  by  25,  and  then 
divide  by  13.  For  f  4  of  95|  is  the  same  as  yL  of  25  times 
951. 

Operation, 

951  X  25  ==:  2396^.       239()|  (13 

13        


109 
104 


18^W. 


52 

^—  \K    Ans.  184-/-\  dolls. 

In  this  example  I  divide  2396|-  by  13.  I  obtain  a  quo- 
tK'nt  184,  and  a  remainder  4|-,  which  is  equal  to  %^.  Then 
Y  divided  by  13,  gives  f-^,  which  I  annex  to  the  quotient, 
and  the  division  is  completed. 

Tlie  examples  hitherto  employed  to  illustrate  the  division 
of  fractions,  have  been  such  as  to  require  the  division  of  the 
fractions  into  parts.  It  has  been  shown  (Art.  XVI.)  that 
the  division  of  whole  numbers  is  performed  in  the  same  man- 
ner, whether  it  be  required  to  divide  the  number  into  parts, 
or  to  fiad  how  many  times  one  number  is  contained  in 
another.  It  will  nov/  be  shown  that  the  same  is  true  with 
regard  to  fractions. 

At  3  dollars  a  barrel,  how  many  barrels  of  cider  may  be 
bought  for  8|  dollars  ? 

The  numbers  must  be  reduced  to  fifths,  for  the  same  rea- 
son that  they  must  be  reduced  to  pence,  if  one  of  the  num- 
bers were  given  in  shillings  and  pence. 

3  =  y ,  and  S-f  =:=  t?.  As  many  times  as  '/  are  contain- 
ed in  Y»  that  is,  as  many  times  as  15  are  contained  in  43,  so 
nany  barrels  may  be  bought. 

Expressing  the  division  4|  =  2||.      Ans.  2||   barrels. 
This  result  agrees  with  the  manner  explained  above.     For 
8^  was  reduced  to  fifths,  and  the  denominator  15  was  formed 
by  multiplying  the  denominator  5  by  the  divisor  3. 
15 


170  ARITHMETIC.  Fart  % 

How  many  times  is  2  contained  in  ^  ? 
2  =  y  ;  14  is  contained  in  5,  -f-^  of  one  time.     The  same 
result  may  be  produced  by  the  other  method. 

XVIII.  We  have  seen  that  a  fraction  may  be  divided  by 
multiplying  its  denominator,  because  the  parts  are  made 
smaller.  On  the  contrary,  a  fraction  may  be  multiplied  by 
dividing  the  denominator,  because  the  parts  will  be  made 
larger.  °  If  the  denominator  be  divided  by  2,  for  instance, 
the^denominator  being  rendered  only  half  as  large,  the  unit 
will  be  divided  into  only  one  half  as  many  parts,  consequently 
the  parts  will  be  twice  as  large  as  before.  If  the  denominator 
be  divided  by  3,  the  unit  will  be  divided  into  only  one  third 
as  many  parts,  consequently  the  parts  will  be  three  times  as 
large  as  before,  and  if  the  same  number  of  these  parts  be 
taken,  the  value  of  the  fraction  will  be  three  times  as  great, 
and  so  on. 

If  1  lb.  of  sugar  cost  i  of  a  dollar,  wJiat  loill  4  Ih.   cost  1 

If  the  denominator  8  be  divided  by  4,  the  fraction  becomes 
\ ;  that  is,  the  dollar,  instead  of  being  divided  into  8  parts, 
is  divided  into  only  2  parts.  It  is  evident  that  halves  are  4 
times  as  large  as  eighths,  because  if  each  half  be  divided  into 
4  parts,  the  parts  will  be  eighths.     Ans.  \  doll. 

If  it  be  done  by  multiplying  the  numerator,  the  answer  is 
f,  which  is  the  same  as  |,  for  |  =  1,  and  i  of  |  =  |. 

If  1  Ih.  of  figs  cost  ^  of  a  dollar,  what  2uill  1  lb.    cost  ? 

Dividing  the  denominator  by  7,  the  fraction  becomes  J. 
Now  it  is  evident  that  fourths  are  7  times  as  large  as  twenty- 
eighths,  because  if  fourths  be  divided  into  7  parts,  the  parts 
wiil  be  twenty-eighths.     Ans.  |  dolls. 

Or  multiplying  the  numerator,  7  times  -^\  is  ||-.  But  i 
r=  -^g-,  and  f  z=z  II-,  so  that  the  answers  are  the  same. 

Therefore,  to  multiply  a  fraction,  divide  the  denominator, 
when  it  can  be  done  ivithout  a  remainder. 

Two  ways  have  now  been  found  to  multiply  fractions,  and 
two  ways  to  divide  them. 

To  nmltiphj  a  fraction  )  "^  r^  f  The  numeimtor,  Art.  15.^ 
To  divide  a  fraction      j  ^,  •§"  \  The  denominator,  Art.  17. 
To  divide  a  fraction      Ki  ^   f  ^'*<^  numerator,  Art.  17. 
To  multiply  a  fraction  j  i^  "g  (  The  denominator,  Art,  18. 


XIX.  DIVISION.  171 

XIX.  We  observed  a  remarkable  circuinaia.ice  in  the 
last  article,  viz.  that  ^  z=  |  and  ^=:  |^.  This  will  be  tbund 
very  important  in  what  follows. 

A  man  liainng  a  cask  of  wine,  sold  i  of  it  at  one  time,  and 
\  of  it  at  another,  how  much  had  he  left  ? 

I  and  \  cannot  be  added  together,  because  the  parts  are 
of  different  values.  Their  sum  must  be  more  than  |,  and 
less  than  |  or  1.  If  we  have  dollars  and  crowns  to  add  to- 
gether, we  reduce  them  both  to  pence.  Let  us  see  if  these 
fractions  cannot  be  reduced  both  to  the  same  denomination. 
Now  i  =  f  =r  I  z=  |,  (fee.  And  J  ==  f  :=  f ,  &.c.  It  ap- 
pears, therefore,  that  they  may  both  be  changed  to  sixths, 
i^  zr:  I  and  -3=  f ,  which  added  together  make  f.  He  had 
£old  I  and  had  {-  left. 

A  man  sold  -|  of  a  barrel  of  four  at  one  time,  and  f  at 
another,  how  much  did  he  sell  in  the  whole  ? 

Fifths  and  sevenths  are  different  parts,  but  if  a  thing  be 
first  divided  into  5  equal  parts,  and  then  those  parts  each  into 
7  equal  parts,  the  parts  will  be  thirty  fifths.  Also  if  the 
thing  be  divided  first  into  7  equal  parts,  and  then  those  parts 
each  into  5  equal  parts,  the  parts  will  be  thirty-fifths. 
Therefore,  the  parts  will  be  alike.  But  in  dividing  them 
thus,  4  w^ill  make  ||,  and  \  will  make  ||-,  and  the  two  added 
together  make  ^^,  that  is,  13^5-.     Ans.  \-^-^  barrel. 

When  the  denominators  of  two  or  more  fractions  are  alike, 
they  are  said  to  have  a  common  denominator.  And  the  pro- 
cess by  which  they  are  made  alike,  is  called  reducing  them 
to  a  common  denominator. 

In  order  to  reduce  pounds  to  shillings,  we  multiply  by  20, 
and  to  reduce  guineas  to  shillings,  we  multiply  by  28.  In 
like  manner  to  reduce  two  or  more  fractions  to  a  common 
denominator,  it  is  necessary  to  find  what  denomination  they 
may  be  reduced  to,  and  \v\\?A  number  the  parts  of  each  must 
be  multiplied  by,  to  reduce  them  to  that  denomination. 

If  the  denominator  of  a  fraction  be  multiplied  by  2,  it  is 
the  same  as  if  each  of  the  parts  were  divided  into  2  equal 
parts,  therefore  it  will  take  2  parts  of  the  latter  kind  to  make 
I  of  the  former.  If  the  denominator  be  multip.'ied  by  8,  it  is 
the  same  as  if  the  parts  were  divided  each  into  3  equal  parts, 
and  it  will  take  3  parts  of  the  latter  kind,  to  make  1  of  the 
former.  Indeed,  whatever  number  the  denominator  be  mul- 
tiplied by,  it  is  the  same  as  if  the  parts  were  each  divided 
into  so  many  equal  parts,  and  it  will  take  so  many  parts  of 


172  ARITHMETIC.  Part  2. 

the  latter  kind  to  make  1  of  the  former.  Therefore,  to  find 
what  the  parts  must  be  multiplied  by,  it  is  necessary  to  find 
what  the  denominator  must  be  muhiplied  by  to  produce  the 
denominator  required. 

The  common  denominator  then,  (which  must  be  found 
first)  must  be  a  number  of  which  tlie  denominators  of  all  the 
fractions  to  be  reduced,  are  factors.  We  shall  always  find 
such  a  number,  by  multiplying  the  denominators  together. 
Hence  if  there  are  only  two  fractions,  the  denominators  be- 
ing multiplied  together  for  the  common  denominator,  the 
parts  of  one  fraction  must  be  multiplied  by  the  denominator 
of  the  other.  If  there  be  more  than  two  fractions,  since  by 
multiplying  all  the  denominators  together,  the  denominator 
of  each  will  be  multiplied  by  all  the  others,  the  parts  in  each 
fraction,  that  is,  the  numerators  must  be  multiplied  by  the 
denominators  of  the  other  fractions. 

In  the  above  example  to  reduce  |  and  -f  to  a  common  de- 
nominator, 7  times  5  are  35  ;  7  is  the  number  by  \\hich  the 
first  denominator  5  must  be  multiplied  to  produce  35,  and 
consequently  the  number  by  which  the  numerator  3  must  be 
multiplied.  5  is  the  number,  by  which  7,  the  second  denomi- 
nator, must  be  multiplied  to  produce  35,  and  consequently 
the  number  by  which  the  numerator  4  must  be  multiplied. 

N.  B.  It  appears  from  the  above  reasoning,  that  if  botli 
the  numerator  and  denominator  of  any  fraction  be  multiplied 
by  the  same  number,  the  value  of  the  fraction  will  remain 
the  same.  It  will  follow  also  from  this,  that  if  both  numera- 
tor and  denominator  can  be  divided  by  the  same  number, 
without  a  remainder,  the  value  of  the  fraction  will  not  be 
altered.  In  fact,  if  the  numerator  be  divided  by  any  num- 
ber, as  3  for  example,  it  is  taking  ^  of  the  number  of  parts  ; 
then  if  the  denominator  be  divided  by  3,  these  parts  will  be 
made  3  times  as  large  as  before,  consequently  the  value  will 
be  the  same  as  at  first.  This  enables  us  frequently,  when  a 
fraction  is  expressed  with  large  numbers,  to  reduce  it,  and 
express  it  with  much  smaller  numbers,  which  often  saves  a 
great  deal  of  labour  in  the  operations. 

Take  for  example  -|f .  Dividing  the  numerator  by  5,  we 
fake  J  of  the  parts,  then  dividing  the  denominator  by  5,  the 
parts  are  made  5  times  as  large,  and  the  fraction  becomes  f , 
the  same  value  as  if.  This  is  called  i^educing  fractions  to 
lower  terms.     Hence 

To  reduce  a  fraction  to  lower  terms,  divide  hath  the  nume- 


XIX.  DIVISION.  17:) 

rator  and  denominator  hy  any  number  that  will  divide  them 
both  without  a  remainder. 

Note.  This  gives  rise  to  a  question,  how  to  find  the  divi- 
sors of  numbers.  These  may  frequently  be  found  by  trial. 
The  question  will  be  examined  hereafter. 

A  man  bought  4  pieces  of  cloth,  the  Jirst  contained  2J3| 
yards ;  the  second  28/2  ;  ^^*c  third  37-j^j  ;  and  the  fourth 
17|.     Hoio  many  yards  in  the  whole  ? 

The  fractional  parts  of  these  numbers  cannot  be  added  to- 
gether until  they  are  reduced  to  a  common  denominator. 
But  before  reducing  them  to  a  common  denominator,  I  ob- 
serve that  some  of  them  may  be  reduced  to  lower  terms, 
which  will  render  it  much  easier  to  find  the  common  denom- 
inator. In  I  the  numerator  and  denominator  may  both  be 
divided  by  2,  and  it  becomes  ^.  -y%  may  be  reduced  to  ^, 
and  y^;^  to  }.  I  find  also  that  halves  may  be  reduced  to 
fourths,  therefore  I  have  only  to  find  the  common  denomina- 
tor of  the  three  first  fractions,  and  the  fourth  can  be  reduced 
to  the  same. 

Multiplying  the  denominators  together  3x4x5  =  60. 
The  common  denominator  is  60.  Now  3  is  multiplied  by 
4  and  by  5  to  make  60,  therefore,  the  numerator  of  |  must 
be  multiplied  by  4  and  by  5,  or,  which  is  the  same  thing,  by 
20,  which  makes  40,  f  z=  ^^.  In  f ,  the  four  is  multiplied 
by  3  and  5  to  make  60,  therefore  these  are  the  numbers  by 
which  the  numerator  3  must  be  multiplied.  ^  =  -J^.  In  the 
fraction  ^,  the  5  is  multiplied  by  3  and  4  to  make  60,  there- 
fore these  are  the  numbers  by  which  the  numerator  1  must 
be  multiplied.  4  =  ^f.  i  z=  -|^.  These  results  may  be  veri- 
fied, by  taking  f,  |,  and  }  of  60.  It  will  be  seen  that  ^  of 
60  is  20,  the  product  of  4  and  5  ;  i  of  60  is  15,  the  product 
of  3  and  5  ;  and  ^  of  60  is  12,  the  product  of  3  and  4 
Now  the  numbers  may  be  added  as  follows : 

23|    =z23f=:23^f  45 

40 
12 
30 

Ans.  107/,-  yards.  127  W  =  Vo- 

I  add  together  the  fractions,  which  make  \y  z=z  2/^.  I 
write  the  fraction  /,,  and  add  the  2  whole  ones  with  the 
others. 

15  • 


28-j-V  =  28|. 

=  2810 

37y3-^37i: 

=  37i§ 

17^    = 

m^ 

174  ARITHMETIC.  Part  2. 

A  man  having  23|  barrels  of  fiour,  sold  8^  barrels  ;  how 
many  barrels  had  he  left  ? 

The  fractions  ~  and  |  must  be  reduced  to  a  comnrion  de- 
nominator, before  the  one  can  be  subtracted  from  the  other. 
I  —  14  and  4  =  ^f     Therefore 
2Sfr=23ii 

But  if  is  larger  than  ^^  and  cannot  be  subtracted  from  it. 
To  avoid  this  diflicuhy,  1  must  be  taken  from  23  and  reduc- 
ed to  21ths,  thus, 

23i-t==22-[-li|  =  22-|f 

Ans.  14|^  yards. 
-|4  taken  from  ||-  leaves  |^.     Then  8  from  22  leaves  14. 
Ans.  14|f  yards. 

From  the  above  examples  it  appears  that  in  order  to  add 
or  subtract  fractio7is,  ichen  they  have  a  common  denominator , 
we  must  add  or  subtract  their  numerators;  and  if  they  have 
not  a  common  denominator,  they  must  first  be  reduced  to  a 
common  denominator. 

We  find  also  the  following  rule  to  reduce  them  to  a  com- 
mon denominator :  midtiply  all  the  denominators  together ^ 
for  a  common  denominator,  and  then  nudtiply  each  numera- 
tor by  all  the  denominators  except  its  oicn. 

XX.  This  seems  a  proper  place  to  introduce  some  con- 
tractions in  division. 

If  24  barrels  of  flour  cost  192  dollars,  7vhat  is  that  a 
barrel  ? 

This  example  may  be  performed  by  short  division.     First 
find  the  price  of  6  barrels,  and  then  of  1  barrel ;  G  ])arrel3 
will  cost  i  of  the  price  of  24  barrels. 
192  (4 


Price  of  6  bar.  48  (6 

Price  of  1  bar.  8  dolls.     Ans. 

If  56 pieces  of  cloth  cost  $7580.72,  tohat  is  it  apiece  ? 
First  find  the  price  of  7,  or  of  8  pieces,  and  then  of  ] 
piece.     7  pieces  v.'ill  cost  -^^  of  the  price  of  B6  pieces. 


XX.  DIVISION. 

7580.72  (8 


Price  of  7  pieces       947.59  (7 


Price  of  1  piece       $135.37     Ans. 

Divide  $24674  equally  ainong  63  men.     How  much  wiU 
each  have  7 

First  find  the  share  of  7  or  9  men,  and  then  of  1  man. 
The  share  of  7  men  will  be  ^  of  the  whole.     The  share  of  9 


will  be  4  of  the  whole. 

24674  (9 

Share  of  7  men 

274If  (7 

Share  of  1  man 

8391|i    Ans. 

24674  (7 

Share  of  9 

men         3524f-  (9 

Share  of  1 

man         8391|J-    Ans. 

In  the  first  case  I  divide  by  9,  and  then  by  7.  Jo  dividmg  hj 
7  there  is  a  remainder  of  4|-,  which  is  V  ;  this  divided  by  7 
gives  ||.  In  the  second  case,  I  divide  by  7  and  then  by  9. 
In  dividing  by  9  there  is  a  remainder  of  5f,  which  is  ^  » 
this  divided  by  9  giVes  |i  as  before. 

Divide  75345  dollars  equally  among  1800  men^  how  much 
will  each  have  1 

First  find  the  share  of  18  men,  which  will  be  ^\-^  part  of  the 
whole.  y|^  part  is  found  by  cutting  off  the  two  right  hand 
figures  and  making  them  the  numerator  of  a  fraction,  thus, 
753,V^. 

Share  of  18  men  $753yVo^  (IS 
72  — 

—  $4imf  Ans.  share  of  1  man. 

33 

18 

153-VV  =  VoV  ;  this  divided  by  IS  is -1^^. 


176  ARITHMETIC.  Parfl 

It  may  be  done  as  follows  : 

Share  of  18  men     753yVo  (^ 


Share  of  3  men       125f  ^  (3 


Share  of  1  man       $41jf  |^-  Ans. 

In  the  last  case  I  find  the  share  of  3  men,  and  then  of  1 
man.  In  dividing  by  6  there  is  a  remainder  3^/^^,  which  is 
^A5^  this  divided  by  6  gives  a  fraction  |^f .  In  dividing  by 
3  there  is  a  remainder  S-gff,  which  is  equal  to  \%\f ,  this  di- 
vided by  3  gives  a  fraction  jf^f ,  and  the  answer  is  $41  j|-4l| 
each. 

From  these  examples  we  derive  the  following  rule  :  When 
the  divisor  is  a  compound  numher,  separate  the  diinsor  into 
two  or  more  factors,  and  divide  the  dividend  by  one  factor  of 
the  divisor^  and  that  quotient  hy  another,  and  so  on,  until 
you  have  divided  hy  the  whole,  and  the  last  quotient  will  be 
the  quotient  required. 

When  there  are  zeros  at  the  right  of  the  divisor,  you  may 
cut  them  off,  and  as  many  figures  from  the  right  of  the  divi- 
dend, making  the  figures  so  cut  off  the  numerator  of  a  frac- 
tion, and  1  with  the  zeros  cut  off,  will  be  the  denominator ; 
then  divide  by  the  remaining  figures  of  the  divisor. 

XXI.  In  Art.  XIX,  it  was  observed,  that  if  both  the  nu- 
merator and  denominator  of  a  fraction  can  be  divided  by  the 
same  number,  without  a  remainder,  it  may  be  done,  and  the 
value  of  the  fraction  will  remain  the  same.  This  gives  rise 
to  a  question,  how  to  find  the  divisors  of  numbers. 

It  is  evident  that  if  one  number  contain  another  a  certain 
number  of  times,  twice  that  number  will  contain  the  other 
twice  as  many  times  ;  three  times  that  number  will  contain 
the  other  thrice  as  many  times,  &c.  that  if  one  number  is 
divisible  by  another,  that  number  taken  any  number  of 
times  will  be  divisible  by  it  also. 

10  (and  consequently  any  number  of  tens)  is  divisible  by 
2,  5,  and  10  ;  therefore  if  the  right  hand  figure  of  any  num- 
ber is  zero,  the  number  may  be  divided  by  either  2,  5,  or  10. 
If  the  right  hand  figure  is  divisible  by  2,  the  number  may  be 
divided  by  2.  If  the  right  hand  figure  is  5,  the  number  may 
be  divided  by  5. 

100  (and  consequently  any  number  of  hundreds)  is  divisi- 


XXI.  DIVISION.  177 

ble  by  4  ;  therefore  if  the  two  right  liand  figures  taken  to- 
gether  are  divisible  by  4,  the  number  may  be  divided  by  4. 

200  is  divisible  by  S  ;  therefore  if  the  hundreds  are  even, 
and  the  tu-o  right  hand  figures  are  divisible  by  8,  the  number 
may  be  divided  by  8.  But  if  the  hundreds  are  odd,  it  will  be 
necessary  to  try  the  three  right  hand  figures.  1000,  being 
even  hundreds,  is  divisible  by  8. 

To  find  if  a  number  is  divisible  by  3  or  9,  add  together  all 
the  figures  of  the  number,  as  if  they  were  units,  and  if  the 
sum  is  divisible  by  3  or  9,  the  number  may  be  divided  by  tl 
or  9. 

The  number  387  is  divisible  by  3  or  9,  because  3  -j-  B 
-|-  7  zn  IS,  which  is  divisible  by  both  3  or  9. 

The  proof  of  the  above  rule  is  as  follows :  10  =:  9  -j-  1  ; 
20  =  2x9-1-2;  30  =  3x9  +  3;  52  =  5  X  9 -f- 5 -f 
2;  100  =  99  +  1;  200=2  X  99  +  2;  387  =  3  X  99  + 
3  +  8x9  +  8  +  7  =  3x99  +  8x9  +  3  +  8  +  7. 
That  is,  in  all  cases,  if  a  number  of  tens  be  divided  by  9,  the 
remainder  will  be  equal  to  the  number  of  tens  ;  and  if  a 
number  of  hundreds  be  divided  by  9,  the  remainder  will 
always  be  equal  to  the  number  of  hundreds.  The  same  is 
true  of  thousands  and  higher  numbers.  Therefore,  if  the 
tens,  hundreds,  thousands,  &-c.  of  any  number  be  divided 
separately  by  9,  the  remainders  will  be  the  figures  of  that 
number,  as  in  the  above  example  387.  Now  if  the  sum  of 
these  remainders  be  divisible  by  9,  the  whole  number  must 
be  so.  But  as  far  as  the  number  may  be  divided  by  9,  it 
may  be  divided  by  3  ;  therefore,  if  the  sum  of  the  remain- 
ders, after  dividing  by  9,  that  is,  the  sum  of  the  figures  are 
divisible  by  3,  the  whole  number  will  be  divisible  by  3. 

The  numbers  G15,  156,  3846,  28572  are  divisible  by  3, 
because  the  sum  of  the  figures  in  the  first  is  12,  in  the  sec- 
ond 12,  in  the  third  21,  and  in  the  fourth  24. 

The  numbers  216,  378,  6453,  and  804672  are  divisible 
by  9,  because  the  sum  of  tlie  figures  in  the  first  is  9,  in  the 
second  18,  in  the  third  18,  and  in  the  fourth  27. 

When  a  number  is  divisible  by  both  2  and  3,  it  is  divisible 
by  their  product  6.  If  it  is  divisible  by  4  and  3  or  5  and  3. 
it  is  divisible  by  their  })roducts  12  and  15.  In  fine,  when  a 
number  is  divisible  by  any  two  or  more  numbers,  it  is  divisi- 
ble by  their  product. 

N.  B.  To  know  if  a  number  is  divisible  by  7, 1 1,  23,  &-{!. 
It  must  be  found  by  trial. 


178  ARITHMETIC.  Part  2. 

When  two  or  more  numbers  can  be  divided  by  the  same 
number  without  a  remainder,  that  number  is  called  their 
common  divisor,  and  the  greatest  number  which  will  divide 
them  so,  is  called  their  greatest  common  divisor.  When  two 
or  more  numbers  have  several  common  divisors,  it  is  evident 
that  the  greatest  common  divisor  will  be  the  product  of  ihem 
all. 

In  order  to  reduce  a  fraction  to  the  lowest  terms  possible, 
it  is  necessary  to  divide  the  numerator  and  denominator  b 
all  their  common  divisors,  or  by  their  greatest  common  divi 
sor  at  first. 

Reduce  ^f  |  to  its  lowest  terms. 

I  observe  in  the  first  place  that  both  numerator  and  de- 
nominator are  divisible  by  9,  because  the  sum  of  the  figures 
in  each  is  9.  I  observe  also,  that  both  are  divisible  by  2, 
because  the  right  hand  figure  of  each  is  so  ;  therefore  they 
are  both  divisible  by  18.  But  it  is  most  convenient  to  divide 
by  them  separately. 

342    V*^  3  8    V'-  19* 

7  and  19  have  no  common  divisor,^  therefore  /tt  cannot  be 
reduced  to  lower  terms. 

The  greatest  common  divisor  cannot  always  be  found  by 
the  above  method.  It  will  therefore  be  useful  to  find  a  rule 
by  which  it  may  always  be  discovered. 

Let  us  take  the  same  numbers  126  and  342. 

126  is  a  number  of  even  18s,  and  342  is  a  number  of  even 
18s ;  therefore  if  126  be  subtracted  from  342,  the  remainder 
216  must  be  a  number  of  even  18s.  And  if  126  be  sub- 
tracted from  216,  the  remainder  90  must  be  a  number  of 
even  18s.  Now  I  cannot  subtract  126  from  90,  but  since  90 
is  a  number  of  even  ISs,  if  I  subtract  it  from  126,  the  re- 
mainder 36  must  be  a  number  of  even  18s.  Now  if  36  be 
subtracted  from  90,  the  remainder  54  must  be  a  number  of 
even  18s.  Subtracting  36  from  54,  the  remainder  is  18. 
Thus  by  subtracting  one  number  from  the  other,  a  smaller 
number  was  obtained  every  time,  which  was  always  a  num- 
ber of  even  18s,  until  at  last  I  came  to  18  itself.  If  18  be 
subtracted  twice  from  36  there  will  be  no  remainder.  It  is 
easy  to  see,  that  whatever  be  tlie  common  divisor,  since  each 
number  is  a  certain  number  of  times  the  common  divisor,  if 
one  be  subtracted  from  the  other,  the  remainder  will  be  a 
certain  number  of  times  the  common  divisor,  that  is,  it  will 
have  the  same  divisor  as  the  numbers  themselves     And  every 


XXI.  DIVISION.  179 

time  the  subtraction  is  made,  a  new  number,  smaller  than 
the  last,  is  obtained,  which  has  the  same  divisor  ;  and  at 
length  the  remainder  must  be  the  common  divisor  itself; 
and'if  this  be  subtracted  from  the  last  smaller  number  as 
many  times  as  it  can  be,  there  will  be  no  remainder.  By 
this  it  may  be  known  when  the  common  divisor  is  found. 
It  is  the  number  which  being  subtracted  leaves  no  remainder. 

When  one  number  is  considerably  larger  than  the  other, 
division  may  be  substituted  for  subtraction.  The  remainders 
only  are  to  be  noticed,  no  regard  is  to  be  paid  to  the  quo- 
tient. 

Reduce  the  fraction  f||  to  its  lowest  terms. 

Subtracting  330  from  462,  there  remains  132.  132  may 
be  subtracted  twice,  or  which  is  the  same  thing,  is  contained 
twice  in  330,  and  there  is  G6  remainder.  ()6  may  be  sub- 
tracted twice  from  132,  or  it  is  contained  twice  in  132,  and 
leaves  no  remainder  ;  66  therefore  is  the  greatest  common 
divisor.  Dividing  both  numerator  and  denominator  by  66, 
the  fraction  is  reduced  to  -f. 

Operation. 
462  (330  330  (66  =  4 

330    

1  462 

330  (132 

264  

2 

132  (66 
132  — 
2 

From  the  above  examples  is  derived  the  following  general 
rule,  to  find  the  greatest  common  divisor  of  two  numbers  : 
Divide  the  greater  hy  the  less,  and  if  there  is  no  remainder, 
that  number  is  itself  the  divisor  required ;  but  if  there  is  a 
remainder,  divide  the  divisor  by  the  remainder,  and  then  di- 
vide the  last  divisor  by  that  remainder,  and  so  on,  until  there 
is  no  remainder,  and  the  last  divisor  is  the  divisor  required. 

If  there  be  more  than  two  numbers  of  which  the  greatest 
common  divisor  is  to  be  found ;  find  the  greatest  common  di- 
visor of  two  of  them,  and  then  take  that  common  divisor  and 
one  of  the  other  numbers,  and  find  their  greate<it  common  di- 
visor  and.  ^o  nn 


180  ARri'HMETIC.  Part  2. 

Reduce  the  fraction  j\  to  its  lowest  terms. 
17  (9 
9    - 
—    1 
9  (8  1  is  the  greatest  common  divisor  iii 

8    -        this   example.      Therefore  the    fraction 
—    1         cannot  be  reduced. 

1(1 
1  - 
-  1 
0 

XXII.  The  method  for  finding  the  common  denomina- 
tor, given  in  Art.  XIX.  though  always  certain,  is  not  always 
the  best ;  for  it  frequently  happens  that  they  may  be  reduced 
to  a  common  denominator,  much  smaller  than  the  one  obtain- 
ed by  that  rule. 

Reduce  |-  and  ^  to  a  common  denominator. 

According  to  the  rule  in  Art.  XIX.,  the  common  denomi- 
nator will  be  54,  and  f  =  |f  and  f  n:  -L|. 

It  was  observed  Art.  XIX.,  that  the  common  denominator 
may  be  any  number,  of  which  all  the  denominators  are  fac- 
tors. 6  and  9  are  both  factors  of  18,  therefore  they  may  be 
both  reduced  to  18ths,|-  =  ||,  and  f  :=  j\. 

When  the  fractions  consist  of  small  numbers,  the  least 
denominator  to  which  the  fractions  can  be  reduced,  may  be 
easily  discovered  by  trial  ;  but  when  they  are  large  it  is  more 
difficult.     It  will,  therefore,  be  useful  to  find  a  rule  for  it. 

Any  number,  which  is  composed  of  two  or  more  factors, 
is  called  a  multiple  of  any  one  of  those  factors.  Thus  IS  is 
a  multiple  of  2,  or  of  3,  or  of  6,  or  of  0.  It  i^  also  a  com- 
mon multiple  of  tht-be  numbers,  that  is,  It  may  be  produced 
by  multiplying  either  of  them  by  some  number.  The  least 
common  multiple  of  two  or  more  numbers,  is  the  least  num- 
ber of  which  they  are  all  factors.  54  is  a  common  multiple 
of  6  and  9,  but  their  least  common  multiple  is  18. 

The  least  common  denominator  of  (vvo  oi  inuie  Iraciious 
will  be  the  least  common  multiple  of  all  the  denominators ; 
the  fraptions  being  previously  reduced  to  their  lowest  terms. 
One  number  will  always  bo,  a  multiple  of  another,  when 
the  former  contains  all  the  factors  of  the  latter.  6  =  2  X  3> 
and  9  =  3  X  3,  and  18  =  2  X  3  X  3.  IS  contains  the  Vic- 
tors 2  and  3  of  6  and  3  and  3  of  9      54  —  2  X  S  X  3  X  3. 


XXIII.  DIVISION.  181 

54,  which  is  produced  by  multiplyinjr  6  and  9,  contains  all 
these  factors,  and  one  of  them,  viz.  JJ,  repeated.  Tlie  rea- 
son why  3  is  repeated  is  because  it  is  a  factor  of  both  G  and 
9.  By  reason  of  this  repetition,  a  number  is  produced  3 
times  as  large  as  is  necessary  for  the  common  multiple. 

When  the  least  common  multiple  of  two  or  more  numbers 
is  to  be  found,  if  two  or  more  of  them  have  a  common  fac- 
tor, it  may  be  left  out  of  all  but  one,  because  it  will  be  suffi- 
cient that  it  enters  once  into  the  product. 

These  factors  will  enter  once  into  the  product,  and  only 
once,  if  all  the  numbers  ichicli  have  common  factors  he  divid- 
ed hy  those  factors  ;  and  then  the  undivided  numbers,  and 
these  quotients  be  multiplied  together,  and  the  product  mul- 
tiplied by  the  common  factors. 

If  any  of  the  quotients  be  found  to  have  a  common  factor 
urith  either  of  the  numbers,  or  with  each  other,  they  may  he 
divided  by  that  also. 

Reduce  f ,  |,  |,  and  |,  to  the  least  common  denominator. 

The  least  common  denominator  will  be  the  least  common 
multiple  of  4,  9,  6,  and  5. 

Divide  4  and  6  by  2,  the  quotients  are  2  and  3.  Then 
divide  3  and  9  by  3,  the  quotients  are  1  and  3.  Then  mul- 
tiplj  '.ng  these  quotients,  and  the  undivided  number  5,  we 
have  2  X  1  X  3  X  5  =  30.  Then  multiplying  30  by  the 
two  common  factors  2  and  3,  we  have  30  X  2  X  3  =  180, 
which  is  to  be  the  common  denominator. 

Now  to  find  how  many  ISOths  each  fraction  is,  take  J,  |-, 
|,  and  I  of  180.  Or  observe  the  factors  of  which  180  was 
made  up  in  the  multiplication  above.  Thus  2  X  1  X  3  X 
•'5x2x3=  180.  Then  multiply  the  numerator  of  each 
fraction  by  the  numbers  by  which  the  fectors  of  its  denomi- 
nator were  multiplied. 

The  factors  2  and  2  of  the  dei.ominator  of  the  first  frac- 
tion, were  multiplied  by  1,  3,  3,  and  5.  The  factors  3  and 
3,  of  the  second,  were  multiplied  by  2,  1,  5,  and  2.  The 
factors  2  and  3,  of  the  third,  were  multiplied  by  2.  1,  3,  5; 
and  5,  the  denominator  of  the  fourth,  was  mult:^>lied  by  2, 
2,  1,  3,  and  3. 

r?  l.'?5.     2  40    .     5   J^  5  0   .     4  14  4 

4  —  TTO   '    "S- T8  9"  '     6"  18  0    ')     5   T8  0* 

XXIII.  If  a  horse  loill  eat  ^  of  a  ushel  of  oats  in  a 
day,  hoio  long  will  12  bushels  last  him  ? 

In  this  question  it  is  required  to  find  how  many  times  J 
16 


182  ARITHMETIC.  Part  2. 

of  a  bushel  is  contained  in  12  bushels.     In  12  there  are  ^«, 
therefore  12  bushels  will  last  36  days. 

At  ^  of  a  dollar  a  busheL,  how  many  bushels  of  corn  may 
be  bought  for  15  dollars  ? 

First  find  how  many  bushels  might  be  bought  at  j  of  a 
dollar  a  bushel.  It  is  evident,  that  each  dollar  would  buy  5 
bushels  :  iherefore  15  dollars  would  buy  15  times  5,  that  is, 
75  buahels.  But  since  it  is  |  instead  of  i  of  a  dollar  a 
bushel,  it  will  buy  only  ^  as  much,  that  is,  25  bushels. 

This  question  is  to  find  how  many  times  f  of  a  dollar,  are 
contained  in  15  dollars.  It  is  evident,  that  15  must  be  reduo- 
ed  to  5ths,  and  then  divided  by  3. 
15 
5 

75  (3 

25  bushels. 
The  above  question  is  on  the  same  principle  as  the  fol- 
lowing. 

How  much  corn,  at  5  shillings  a  bushel,  may  be  bought  for 
23  dollars  ? 

The  dollars  in  this  example  must  be  reduced  to  shillings, 
before  we  can  find  how  many  times  5  shillings  are  contain- 
ed in  them  ;  that  is,  they  must  be  reduced  to  6ths,  before  we 
can  find  how  many  times  ^  are  contained  in  them. 
23 
6 

138  (5 

Ans.  27f  bushels. 
23  z=  1 J8  and  f  are  contained  27|  times  in  *|®. 
If  1^  yds.  of  cloth  will  make  1  suit  of  clothes^  how  many 
suits  will  48  yards  make  1 

If  the  question  was  given  in  yards  and  quarters,  it  is  evi- 
dent both  numbers  must  be  reduced  to  quarters.     In  this 
instance  then,  they  must  be  reduced  to  8ihs. 
7|  —  V  and  48  —  ^f* 
384  (59 
354    — 

6f|  suits.    Ans. 

30 


XXIII.  DIVISION.  183 

In  tlie  three  last  examples,  the  purpose  is  to  find  how 
many  times  a  fraction  is  contained  in  a  whole  number.  This 
is  dividing  a  whole  number  by  a  fraction,  for  which  we  find 
the  following  rule  :  Reduce  the  dividend  to  the  same  denomi- 
nation as  the  divisor,  and  then  divide  hy  the  numerator  of 
the  fraction. 

Note.  If  the  divisor  is  a  mixed  number,  it  must  be  re- 
duced to  an  improper  fraction. 

N.  B.  The  above  rule  amounts  to  this ;  multiply  the  div- 
idend by  the  denominator  of  the  divisor^  and  then  divide  it 
by  the  numerator. 

At  I  of  a  dollar  a  bushel,  how  many  bushels  of  potatoes 
may  be  bought  for  ^  of  a  dollar. 

I  is  contained  in  ^  as  many  times  as  1  is  contained  in  3. 
Ans.  3  bushels. 

Jf  _3^.  of  a  ton  of  hay  will  keep  a  horse  1  mouthy  how 
many  horses  loill  -f^  of  a  ton  keep  the  same  time  ? 

-j?^  are  contained  in  -^^  as  many  times  as  3  are  contained 
in  9.     Ans.  3  horses. 

At  \  of  a  dollar  a  pound,  hoio  many  pounds  of  fgs  may 
be  bought  for  ^  of  a  dollar  ? 

5ths  and  4ths  are  different  denominations  ;  before  one  can 
be  divided  by  the  other,  they  must  be  reduced  to  the  same 
denomination  ;  that  is,  reduced  to  a  common  denominator. 

i-  =  /o  ^"^  I  =  M-  2^0  ^r®  contained  in  ^^  as  many 
times  as  4  are  contained  in  1-5.     Ans.  3|  lb. 

At  7^  dolls,  a  yard,  how  many  yards  of  cloth  may  be 
bought  for  57|-  dollars  1 

7|  z=:  V  and  57|  =  ^|' .  5ths  and  8ths  are  different  de- 
nominations ;  they  must,  therefore,  be  reduced  to  a  common 
denominator. 

38   —   304   nnH    451   —  2-^05 

2305  (304 

2128    

nn  yards. 

177 
From  the  above  examples  we  deduce  the  following  rule, 
ibr  dividing  one  fraction  by  another : 

If  the  fractions  are  of  the  same  denomination,  divide  the 
numerator  of  the  dividend  by  the  numerator  oj  the  divisor. 

If  the  fractions  arc  of  different  denominations,  they  must 
first  be  reduced  to  a  common  denominator. 


184  ARITHMETIC.  Part  2. 

If  eithet  or  both  of  the  numbers  are  mixed  numbers,  they 
must  Jirst  be  reduced  to  improper  fractions. 

Note,  As  the  common  denominator  itself  is  not  used  in 
the  operation,  it  is  not  necessary  actually  to  tind  it,  but  only 
to  multiply  the  numerators  by  the  proper  numbers  to  reduce 
them.  By  examining  the  above  examples,  it  will  be  found 
that  this  purpose  is  effected,  by  multiplying  the  numerator 
of  the  dividend  by  the  denominator  of  the  divisor,  and  the 
denominator  of  the  dividend  by  the  numerator  of  the  divisor. 
Thus  in  the  third  example  ;  multiplying  the  numerator  of  | 
by  5  and  the  denominator  by  1,  it  becomes  '■£ ,  which  reduc- 
ed is  "Sj;  pounds  as  before. 

XXIV.  A  owned  }  of  a  ticket,  ichich  drew  a  prize.  A's  share 
of  the  money  was  567  dollars.    What  was  the  whole  prize  ? 

^  of  a  number  make  the  whole  number.     Therefore  the 
whole  prize  was  5  times  A's  share. 
567 
5 

Ans.  2835  dollars. 
A  man  bought  \  of  a  ton  of  iron  for  13f  dollars  ;  what 
was  it  a  ton  ? 

I  make  the  whole,  therefore  the  whole  ton  cost  7  times  13f . 
13| 
7 

Ans.  95|  dolls. 
A  man  bought  -^^  of  a  ton  of  iron  for  40  dollars ;  what 
was  it  a  ton  ? 

y^Tj-  are  5  times  as  much  as  -f^.  If  ^^  cost  40  dollars,  y^ 
must  cost  \  of  40.  \  of  40  is  8,  and  8  is  yV  o^  ^^-  -^"^• 
96  dollars. 

A  man  bought  j  of  a  ton  of  hay  for  17  dollars ;  what 
was  it  a  ton  1 

f  are  3  times  as  much  as  \.  Smce  4  cost  17  dollars,  \ 
must  cost  ^  of  17,  and  -f  must  cost  f  of  17. 

17  (3     or  multiplying  first  by     17 
the  denominator  5 

5!  - 

5  85  (3 

Ans.  28i  dolls.  28i  dolls. 


XXIV.  DIVISION.  ia5 

If  4|  firkins  of  butter  cost  33  dollars,  what  is  that  a  fir- 
kin ? 

4|  =  ^-^.  First  we  must  find  what  ^  costs.  |  is  jV  pari 
of  V»  therefore  ^  will  cost  gV  of  33  dollars,  and  j^  will  cost 
A  of  33  dollars. 

33 
5 

165  (22 
154   — 

7^  =  7i  dollars 

11 

The  six  last  examples  are  evidently  of  the  same  kind.  In 
al!  of  them  a  part  or  several  parts  of  a  number  were  given  to 
find  the  whole  number.  They  are  exactly  the  reverse  of  the 
examples  in  Art.  XVI.  If  we  examine  them  still  farther, 
we  shall  find  them  to  be  division.  In  the  last  example,  if  4 
firkins  instead  of  4|  had  been  given,  it  would  evidently  be 
division  ;  as  it  is,  the  principle  is  the  same.  It  is  therefore 
dividing  a  whole  number  by  a  fraction  ;  the  general  rule  is, 
to  find  the  value  of  one  part,  and  then  of  the  whole.  To 
find  the  value  of  one  part,  divide  the  dividend  by  the  mime' 
rotor  of  the  divisor ;  and  then  to  find  the  whole  number^ 
multiply  the  part  by  the  denominator. 

Or,  according  to  the  two  last  examples,  midtiply  the  divi- 
dend by  the  denominator  of  the  divisor,  and  divide  by  the 
numerator. 

N.  B.  This  last  rule  is  the  same  as  that  in  Art.  XXIII. 
This  also  shows  this  operation  to  be  division^ 

Note.  If  the  divisor  is  a  mixed  number,  reduce  it  to  an 
improper  fraction. 

If  ^  of  a  yard  of  cloth  cost  j  of  a  dollar,  what  will  a 
yard  0)st  ? 

It  is  evident  that  the  whole  yard  will  cost  5  times  f ,  which 
is  y  =  2f  dollars. 

If  ^  of  a  yard  of  cloth  cost  ^  of  a  dollar,  what  is  that  a 
yard  ? 

If  ^  cost  I,  4  must  cost  ^  of  f  ;  -j  of  f  is  ^5_ .  ^^  being 
^,  7  times  ^^  or  ff  =  IH  dollars  must  be  the  price  of  a 
yard. 

16* 


186  ARITHMETIC.  Fart  2. 

If  3f  barrels  of  flour  cost  23  f  dollars,  ivhat  is  thai  a 
barrel  ? 

^  =1  2_?  and  23f  =  '4^.  If  \^  of  a  barrel  cost  ^^  of 
a  dollar,  }  of  a  barrel  will  cost  2^'9-  of  ^^-^,  2T  of  'f-^  is 
|.6|.  i.6_3  being  1^  of  the  price  of  1  barrel,  8  times  |^|-|  will 
be  the  price  of  a  barrel.  8  times  -^fj  =  'gW  =  ^T^i  ^^^' 
lars.     Ans.  6/^  dollars  per  barrel. 

The  three  last  examples  are  of  the  same  kind  as  those 
which  precede  them  ;  the  only  difference  is,  that  in  these, 
the  part  which  is  given,  or  the  dividend,  is  a  fraction  or 
mixed  number. 

In  this  case  the  dividend,  if  a  mixed  number,  must  be  re- 
duced to  an  improper  fraction  ;  then  in  order  to  divide  the 
dividend  by  the  numerator  of  the  divisor,  it  will  generally  be 
necessary  to  multiply  the  denominator  of  the  dividend  by 
the  numerator  of  the  divisor. 

From  this  article  and  the  preceding,  we  derive  the  follow- 
ing general  rule,  to  divide  by  a  fraction,  whether  the  di- 
vidend be  a  whole  number  or  not :  Multiply  the  dividend  by 
the  denominator  of  the  divisor,  and  divide  the  product  by  the 
7iumerator.  If  the  divisor  is  a  mixed  number,  it  must  be 
changed  to  an  improper  fraction. 


DECIMAL  FRACTIONS, 

XXV.  We  have  seen  that  the  nine,  digits  may  be  made 
to  express  different  values,  by  putting  them  in  different 
places,  and  that  any  number,  however  large,  may  be  ex- 
pressed by  them.  We  shall  now  see  how  they  may  be  made 
to  express  numbers  less  than  unity,  (that  is,  fractions,)  in  the 
same  manner  as  they  do  those  larger  than  unity. 

Suppose  the  unit  to  be  divided  into  ten  equal  parts. 
These  are  called  tenths,  and  ten  of  them  make  1,  in  ♦he 
same  manner  as  ten  units  make  I  ten,  and  as  ten  tens  ma^e 
1  hundred,  &c.  In  the  common  way,  3  tenths  is  written 
^,  and  47  and  3  tenths  is  written  47-f\.  Now  if  we  assign 
a  place  for  tenths,  as  we  do  for  units,  tens,  &c.  it  is  evident 
that  they  may  be  written  without  the  denominator,  and  the}' 
will  be  always  understood  as  tenths.  It  is  agreed  to  write 
tejiths  at  the  right  hand  of  the  units,  separated  from  them  by 


XXV.  DECIMALS.  JS7 

a  point  (.).  Hitherto  we  have  been  accustomed  to  consider 
the  right  hand  figure  as  expressing  units  ;  we  still  consider 
units  as  the  starting  point,  and  must  therefore  make  a  mark, 
in  order  to  show  which  we  intend  for  units.  Thus  47/-. 
47  signifies  4  tens  and  7  units  ;  then  if  we  wish  to  write  \^, 
we  make  a  point  at  the  right  of  7,  and  then  write  3,  thus,' 
47.3.     This  is  read  forty-seven  and  three  tenths. 

Again,  suppose  each  tenth  to  be  divided  into  ten  equal 
parts :  the  whole  unit  will  then  be  divided  into  one  hundred 
equal  parts.  But  they  were  made  by  dividing  tenths  into 
ten  equal  parts,  therefore  ten  hundredths  will  make  one 
tenth.  Hundredths  then  may  with  propriety  be  written  at 
the  right  of  tenths,  but  there  is  no  need  of  a  mark  to  distin- 
guish these,  for  the  place  of  units  being  the  starting  point, 
'.vhen  that  is  known,  all  the  others  may  be  easily  known. 

7^  is  written  7.04.  83.57  is  read  83  and  j%-  and  j^^, 
or  since  ^V  —  _5_o^  vve  may  read  it  83 /q^,  which  is  a  shorter 
expression. 

Again,  suppose  each  hundredth  to  be  divided  into  ten 
equal  parts  ;  these  will  be  thousandths.  And  since  ten  of 
the  thousandths  make  one  hundredth,  these  may  with  pro- 
priety occupy  the  place  at  the  right  of  the  hundredths,  or  the 
third  place  from  the  units. 

It  is  easy  to  see  that  this  division  may  be  carried  as  far  as 
ye  please.  The  figures  in  each  place  at  the  right,  signifv- 
ing  parts  1  tenth  part  as  large  as  those  in  the  one  at  the  leli 
of  it. 

Beginning  at  the  place  of  units  and  proceeding  towards 
the  left,  the  value  of  the  places  increases  in  a  tenfold  propor- 
tion, and  towards  the  right  it  diminishes  in  a  tenfold  pro- 
portion. 

Fractions  of  this  kind  may  be  written  in  this  manner,  when 
there  are  no  whole  numbers  to  be  written  with  them.  ^ 
for  example  may  be  written  0.4,  or  simply  .4.  j^  may  be 
written  0.03  or  .03.  j%\  may  be  written  .87.  The  point 
always  shows  where  the  decimals  begin.  Since  the  value  of 
a  figure  depends  entirely  upon  the  place  in  which  it  is  writ- 
ten, great  care  must  be  taken  to  put  every  one  in  its  proper 
place. 

Fractions  written  in  this  way  are  called  decimal  fractions^ 
from  the  Latin  word  decern,  which  signifies  ten,  because  they 
increase  and  diminish  in  a  tenfold  proportion. 

It  is  important  to  remark  that  ^  =  t\\  =  -^^  =  tW%, 


188  ARITHMETIC.  ParfZ, 

&c.  and  that-pf^  =  -S^P^.^—^^l^- ^&i^c.  and  -/-^  —  ^o'tt'ott,  con- 
sequently ^V  +  rf 0  H-  ToVo  +  Tolno  =  iVo'A  —  ^>.3572. 
Any  other  numbers  may  be  expressed  in  the  same  manner. 
From  this  it  appears  that  any  decimal  may  be  reduced  to  a 
lower  denomination,  simply  by  annexing  zeros.  Also  any 
number  of  decimal  figures  may  be  read  together  as  whole 
numbers,  giving  the  name  of  the  lowest  denomination  to  the 
whole. 

Thus  0.38752  is  actually  ^%  +  ^i^  +  ^/^  +  -,^f ^^  + 
10 0-00  0.  tjut  It  may  all  be  read  together  tVoWo.  thirty-eight 
thousand,  seven  hundred  and  fifty-two  hundred-thousandths. 
Any  whole  number  may  be  reduced  to  tenths,  hundredths, 
&LC.  by  annexing  zeros.  27  is  270  tenths,  2700  hundredths, 
&c.  consequently  27.35  may  be  read  two  thousand,  seven 
hundred  and  thirty-five  hundredths,  Vo¥.  In  like  manner 
any  whole  number  and  decimal  may  be  read  together,  giving 
it  the  name  of  the  lowest  denomination.  It  is  evident  that  a 
zero  at  the  right  of  decimals  does  not  alter  the  value,  but  a 
zero  at  the  left  diminishes  the  value  tenfold. 

It  is  evident  that  any  decimal  may  be  changed  to  a  com- 
mon fraction,  by  writing  the  denominator,  which  is  always 
understood,  under  the  fraction.  Thus  .75  may  be  written 
/o\,  then  reducing  it  to  its  lowest  terms  it  becomes  f .  The 
denominator  will  always  be  1,  with  as  many  zeros  as  there 
are  decimal  places,  that  is,  one  zero  for  tenths,  two  for 
hundredths  &c. 


XXVI.  DECIMALS.  18!) 

The  following  table  exhibits  the  places  with  their  names, 
as  far  as  ten-millionths,  together  with  some  examples. 


m 


'TS    PT-I      C«      V      M5      O 

3 


OGG.ticSOiSC—    - 

H  K  H  ^  H  S  H  H  ffi  S  H 


6  and  7  tenths  6/o  .   .   .   6  .7  .   . 

44  and  3  hundredths  44y|^  .    .  4  4  .0  3  . 

50  and  64  hundredths  50^Vo  •  •  5  0  .6  4  . 
243  and  87  thousandths  243y|^  .  2  4  3  .0  8  7 
9-247  and  204  thousandths 

9247yVVo  9  2  4  7  .2  0  4 
42  and  7  ten-thousandths 


42 ^ .   .  4  2  .0  0  0  7 


3  and  904  ten-thousandths 

3,1^1^  .  .   .   3  .0  9  0  4  .   .   . 

9  tenths  tV  •   •  •  -^ 

3  thousandths  joV^  ...  .003.... 

29  hundredths  tVo  •    •   •  -^  ^ 

8  hundred-thousandths  tofo  o  o  •   •  •  -00008.. 

67  millionths  to oV^oo"  •   •  •  -000067. 

3064  ten-millionths  to^^IW  •   •   •  -0003064 

In  Federal  money  the  parts  of  a  dollar  are  adapted  to  the 
decimal  division  ot  the  unit.  The  dollar  being  the  unit, 
dimes  are  tenths,  cents  are  hundredths,  and  mills  are  thou- 
sandths. 

For  example,  25  dollars,  8  dimes,  3  cents,  7  mills,  are 
written  $25,837,  that  is,  25^^-^^  dollars. 

XXVI.  A  man  purchased  a  cord  of  wood  for  7  dollars^ 
3  rfmes,  7  cents^  5  milh,  that  is,  87.375  ;  a  gallon  of  molas- 
ses for  80.43  ;  1  Ih.  of  coffee  for  $0.27  ;  a  frkin  of  but- 
ter for  88  ;  a  gallon  of  brandy  for  80.875  ;  and  4  eggs 
fur  8C.03.     How  much  did  they  all  come  to  ' 

It  is  easy  to  see  that  dollars  must  be  added  to  dollars. 


190  ARITHMETIC.  Part  2. 

dimes  to  dunes,  cents  to  cents,  and  mills  to  mills.     They 
may  be  written  down  thus  : 

$7,375 

0.430 

0.270 

8.000 

0.875 

0.030 


Ans.  816.980 

A  man  bought  ^^^  barrels  of  flour  at  one  time,  8yV^  bar- 
rels at  another,  y^-^^  barrel  at  a  third,  and  15/^^^-  at  a 
fourth.     How  many  barrels  did  he  buy  in  the  whole  ? 

These  may  be  written  without  the  denominators,  as  fol 
lows  ;  3.3  barrels,  8.63  barrels,  .873  barrel,  15.784  barrels 
It  is  evident  that  units  must  be  added  to  units,  tenths  to 
tenths,  &LC.  For  this  it  may  be  convenient  to  write  them 
down  so  that  units  may  stand  under  units,  tenths  under  tenths, 
&c.  as  follows  : 
3.3 
8.63 

.873 
15.784 


Ans.  28.587  barrels.     That  is,  2SyVV^  barrels. 

1  say  3  (thousandths)  and  4  (thousandths)  are  7  (thou- 
sandths,) which  I  write  in  the  thousandths'  place.  Then  3 
(hundredths)  and  7  (hundredths)  are  10  (hundredths)  and  8 
(hundredths)  are  18  (hundredths,)  that  is,  1  tenth  and  8 
hundredtlis.  I  reserve  the  1  tenth  and  write  the  8  hun- 
dredths in  the  hundredths*  place.  Then  1  tenth  (which  was 
reserved)  and  3  tenths  are  4  tenths,  and  6  are  10,  and  8  are 
18,  and  7  are  25  (tenths,)  which  are  2  whole  ones  and  5 
tenths.  I  reserve  the  2  and  write  the  5  tenths  in  the  tenths' 
place.  Then  2  (which  were  reserved)  and  3  are  5,  and  8 
are  13,  and  5  are  18,  which  is  1  ten  and  8.  I  write  the  8 
and  carry  the  1  ten  to  the  1  ten,  which  makes  2  tens.  The 
answer  is  28.587  barrels. 

It  appears  that  addition  of  decimals  is  performed  in  pre- 
cisely the  same  manner  as  addition  of  whole  numbers^ 
Care  must  be  taken  to  add  units  to  units,  tenths  to  tenths,  Sfc. 
To  prevent  mistakes  it  will  generally  be  most  convenient  ic 


XXVI.  DECIMALS.  191 

wriie  them,  so  that  units  mai/  statid  under  units,  tenths  under 
tenths,  Sfc. 

It  is  plain  that  the  operations  on  decimal  fractions  are  as 
easy  as  those  on  whole  numbers,  but  fractions  of  this  kind  do 
not  often  occur.  We  shall  now  see  that  common  fractions 
may  be  changed  to  decimals. 

A  merchant  bought  6  pieces  of  cloth  ;  the  first  containing 
14|-  yards,  the  second  37^,  the  third  4^,  the  fourth  17|,  the 
fifth  19^,  and  the  sixth  42  ^|.  How  many  yards  in  the 
whole  ? 

14i 

H 

17^ 

19| 

To  add  these  fractions  together  in  the  common  way,  they 
must  be  reduced  to  a  common  denominator.  But  instead  of 
reducing  them  to  a  common  denominator  in  the  usual  way, 
we  may  reduce  them  to  decimals,  which  is  in  fact  reducing 
them  to  a  common  denominator ;  but  the  denominator  is  of 
a  peculiar  kind. 

i  =  y^o,  f  =  /p.  i  cannot  be  changed  to  tenths,  but  it 
may  be  changed  to  hundredths.  ^  =  y^^,  J  —  y'^*^.  |  can- 
not be  changed  to  hundredths,  but  it  may  be  changed  to 
thousandths.  |  =  yV/o-.  ^f  may  be  reduced  to  hundredths. 
1   —  _  5_  and  Al  —  -6-5- 

3^ 10  0»   •*""    2  0   10  0* 

Writing  the  fractions  now  without  their  denominators  m 
the  form  of  decimals,  they  become 
14.5 
37.6 
4.25 
17.75 
19.375 
42.65 


Ans.  136.125  yards  or  136,^7  =  1361  yards. 

Common  fractions  cannot  always  be  changed  to  decimals 
BO  easily  as  those  in  the  above  example,  but  since  there  will 
be  frequent  occasion  to  change  them,  it  is  necessary  to  find 
a  principle,  by  which  it  may  always  be  done. 

A  man  divided  5  bushels  of  wheat  equally  among  8  ^er» 
son$  ;  how  much  did  he  give  them  apiece  ? 


192  ARITHMETIC.  Part  2. 

He  gave  them  |  of  a  bushel  apiece,  expressed  in  the  form 
of  common  fractions ;  but  it  is  proposed  to  express  it  in  de- 
cimals. 

1  first  suppose  each  bushel  to  be  divided  into  10  equal 
parts  or  tenths.  The  five  bushels  make  \^.  I  perceive  that 
I  cannot  divide  |^  into  exactly  8  parts,  therefore  I  suppose 
each  of  these  parts  to  be  divided  into  10  equal  parts  ;  these 
parts  will  be  hundredths.  5  =z  f  |^.  But  500  cannot  be  di- 
vided by  8  exactly,  therefore  I  suppose  these  parts  to  be 
divided  again  into  10  parts  each.  These  parts  will  be  thou- 
sandths. 5  —  f i^^.  5000  may  be  divided  by  8  exactly,  \- 
of  f  0^1  ^s  f  oVo ,  or  .(v25.     Ans.  .625  of  a  bushel  each. 

Instead  of  trying  until  I  find  a  number  that  may  be  ex- 
actly divided,  I  can  perform  the  work  as  I  make  the  trials. 
For  instance,  I  say  5  bushels  are  equal  to  f  ^  of  a  bushel.  ^ 
of  \^Q  is  j^fTj  and  there  are  -^^  left  to  be  divided  into  8  parts. 
I  then  suppose  these  2  tenths  to  be  divided  into  ten  equal 
parts  each.  They  will  make  20  parts,  and  the  parts  are 
hundredths,  i  of  ^-^-^  are  y|^,  and  there  are  y4_  left  to  be 
divided  into  8  parts.  I  suppose  these  4  hundredths  to  l>e 
divided  into  10  parts  each.  They  will  make  40  parts,  and 
the  parts  will  be  thousandths,  -l  of  y^^  is  j-i^Q.  Bringing 
the  parts  j%,  -j^o,  and  jJ^  together,  they  make  jV/^or  .625 
of  a  bushel  each,  as  before. 

The  operation  may  be  performed  as  follows  : 
50  (8 

48  

.625 

20 
16 

40 
40 


I  write  the  5  as  a  dividend  and  the  8  as  a  divisor.  Then 
I  multiply  5  by  10,  (that  is,  I  apnex  a  zero)  in  order  to  re- 
duce the  5  to  tenths.  Then  \  of  50  is  6,  which  I  write  in 
the  quotient  and  place  a  point  before  it,  because  it  is  tenths. 
There  is  2  remainder.  I  multiply  the  2  by  10,  in  order  to 
reduce  it  to  hundredths.  \  of  20  is  2,  and  there  is  4  re- 
mainder.    I  multiply  the  4  by  10,  in  order  to  reduce  it  to 


XXVI.  DECIMALS.  193 

thousandths.     ^  of  40  is  5.     The  answer  is  .625  bushels 
each,  as  before. 

In  Art.  X.  it  was  shown,  that  when  there  is  a  remainder 
after  division,  in  order  to  complete  the  quotient,  it  must  be 
written  over  thf  divisor,  and  annexed  to  the  quotient.  This 
traction  may  be  reduced  to  a  decimal,  by  annexing  zeros, 
ind  continuing  the  division. 
Divide  57  barrels  of  flour  equally  among  16  men, 
57  (16 

48  

3.5625  barrels  each. 

90 
80 

100 
96 

40 
32 

80 
80 


In  this  example  the  answer,  according  to  Art.  X.,  is  3^ 
bushels.  But  instead  of  expressing  it  so,  I  annex  a  zero  to 
the  remainder  9,  which  reduces  it  to  tenths,  then  dividing,  I 
obtain  5  tenths  to  put  into  the  quotient,  and  I  separate  it 
from  the  3  by  a  point.  There  is  now  a  remainder  10,  which 
I  reduce  to  hundredths,  by  annexing  a  zero.  And  then  I 
divide  again,  and  so  on,  until  there  is  no  remainder. 

The  first  remainder  is  9,  this  is  9  bushels,  which  is  yet  to 
be  divided  among  the  16  persons ;  when  I  annex  a  zero  I 
reduce  it  to  tenths.  The  second  remainder  10  is  so  many 
tenths  of  a  bushel,  which  is  yet  to  be  divided  among  the  16 
persons.  When  I  annex  a  zero  to  this  I  reduce  it  to  hun- 
dredths. The  next  remainder  is  4  hundredths,  which  is  yet 
to  be  divided.  By  annexing  a  zero  to  this  it  is  reduced  to 
thousandths,  and  so  on. 

The  division  in  this  example  stops  at  ten-thousandths  ;  the 
reason  is,  because  10000  is  exactly  divisible  by  16.  If  X 
take  j%  of  i-^§^  I  obtain  ■^^^,  or  .5625,  as  above. 

There  are  many  common  fractions  which  require  so  many 
17 


194  ARITHMETIC-  Pait% 

figures  to  express  their  value  exactly  in  decimals,  as  to 
render  them  very  inconvenient.  There  are  many  also,  the 
value  of  which  cannot  be  exactly  expressed  in  decimals. 
In  most  .calculations,  however,  it  will  be  sufficient  to  use  an 
approximate  value.  The  degree  of  approximation  necessary^ 
must  always  be  determined  by  the  nature  of  the  case.  For 
example,  in  making  out  a  single  sum  of  money,  it  is  consi- 
dered sufficiently  exact  if  it  is  right  within  something  less 
than  1  cent,  that  is,  within  less  than  j^^  of  a  dollar.  But  if 
several  sums  are  to  be  put  together,  or  if  a  sum  is  to  be  mul- 
tiplied, mills  or  thousandths  of  a  dollar  must  be  taken  into 
the  account,  and  sometimes  tenths  of  mills  or  ten-thou- 
sandths. In  general,  in  questions  of  business,  three  or  four 
decimal  places  will  be  sufficiently  exact.  And  even  where 
very  great  exactness  is  required,  it  is  not  very  often  neces- 
sary to  use  more  than  six  or  seven  decimal  places. 

A  merchant  bought  4  pieces  of  cloth  ;  the  jirst  contained 
28|  yards  ;  the  second  34|- ;  the  third  BO^'y  ;  and  the  fourth 
42-^  yards.     How  many  yards  in  the  whole  7 

In  reducing  these  fractions  to  decimals,  they  will  be  suffi- 
ciently exact  if  we  stop  at  hundredths,  since  y^  of  a  yard  is 
only  about  \  of  an  inch. 

30  (5        200  (7         100  (15        700  (9 

.6  .28 -f  .07—  .78  — 

I  is  exactly  .6.  If  we  were  to  continue  the  division  off, 
it  would  be  .28571,  &c. ;  in  fact  it  would  never  terminate; 
but  .28  is  within  about  one  i  of  ^i^  of  ^  yai'd,  therefore 
sufficiently  exact.  -^  is  not  so  much  as  ■^^,  therefore  the 
first  figure  is  in  the  hundredths'  place.  The  true  value  is 
.0G6G,  &c.,  but  because  ^/^^^  is  more  than  ^  of  ji^,  I  call 
it  .07  instead  of  .06.  |-  is  equal  to  .7777,  &c.  This  would 
never  terminate.  Its  value  is  nearer  .78  than  .77,  therefore 
I  use  .78. 

When  the  decimal  used  is  smaller  than  the  true  one,  it  is 
well  to  make  the  mark  -f-  after  it,  to  show  that  something 
more  should  be  added,  as  f  :=  .28  -{—  When  the  fraction  is 
too  large  the  mark  —  should  be  made  to  show  that  some- 
thing should  be  subtracted,  as  ^'j  =r  .07  — . 

Tne  numbers  to  be  added  will  now  stand  thus  : 


XXVI.  DECIMALS.  195 

28|    =28.60 

34f    =34.28  4- 

aOfV  — 30.07— 

42|    =42.78  — 


Ans.  135.75  yards,  or  135//^  =  135|. 
From  the  above  observations  we  obtain  the  following  ge- 
neral rule  for  changing  a  common  fraction  to  a  decimal :  Ati- 
ney  a  zero  to  the  numerator,  and  divide  it  by  the  denomina- 
tor, and  then  if  there  he  a  remainder,  annex  another  zero, 
and  divide  again,  and  so  on,  until  there  is  no  remainder,  or 
until  a  fraction  is  obtained,  tvhich  is  sufficiently  exact  for 
the  purpose  required. 

Note.  When  one  zero  is  annexed,  the  quotient  will  be 
tenths,  when  two  zeros  are  annexed,  the  quotient  will  be 
hundredths,  and  so  on.  Therefore,  \f  when  one  zero  is  an- 
nexed, the  dividend  is  not  so  large  as  the  divisor,  a  zero 
must  be  put  in  the  quolient  with  a  point  before  it,  and  in  the 
same  manner  after  two  or  more  zeros  are  annexed,  if  it  is 
nqt  yet  divisible,  as  many  zeros  must  be  placed  in  the  quo- 
tient. 

I'wo  men  talking  of  their  ages,  oJie  said  he  laas  37^^/7^/3 
years  old,  and  the  other  said  he  was  64||^  years  old.  What 
was  the  difference  of  their  ages  7 

If  it  is  required  to  find  an  answer  withm  1  minute,  it  will 
be  necessary  to  continue  the  decimals  to  seven  p'aces,  for  1 
minute  is  -,^\to  «f  ^  year.  If  tne  answer  is  required  only 
within  hours,  five  places  are  sufficient ;  if  only  withm  days, 
four  places  are  sufficient. 

64|4  f  =  64.8^520000 
37-3JM_7- =r  37.2602313 -I- 


Ans.  27.5917687  years. 

It  is  evident  that  units  must  be  subtracted  from  units, 
enths  from  tenths,  &c.  If  the  decmial  places  in  the  two 
numbers  are  not  alike,  they  may  be  made  alike  by  annexing 
zeros.  After  the  numbers  are  prepared,  subtraction  is  per- 
formed precisely  as  in  whole  numbers. 


196  •        ARITHMETIC.  Part% 

Multiplication  of  Decimals, 

XXVII.     How  many  yards  of  cloth  are  there  in  seven 
pieces,  each  piece  containing  19|^  yards  ? 
19^  =:  19.875 

7 


Ans.  139.125  =z  139^^=  139^  yards. 

N.  B.  All  the  operations  on  decimals  are  performed  in 
precisely  the  same  manner  as  whole  numbers.  All  the  diffi- 
culty consists  in  finding  where  the  separatrix,  or  decimal 
point,  is  to  be  placed.  This  is  of  the  utmost  importance, 
since  if  an  error  of  a  single  place  be  made  in  this,  their  value 
is  rendered  ten  times  too  large  or  ten  times  too  small.  The 
purpose  of  this  article  and  the  next  is  to  show  where  the 
point  must  be  placed  in  multiplying  and  dividing. 

In  the  above  example  there  are  decimals  in  the  multipli- 
cand, but  none  in  the  multiplier.  It  is  evident  from  what 
we  have  seen  in  adding  and  subtracting  decimajs,  that  in 
this  case  there  must  be  as  many  decimal  places  in  the  pro- 
duct, as  there  are  in  the  multiplicand.  It  may  perhaps  be 
more  satisfactory  if  we  analyze  it. 

7  times  5  thousandths  are  35  thousandths,  that  is,  3  hun- 
dredths and  5  thousandths.  Reserving  the  hundredths,  I 
write  the  5  thousandths.  Then  7  times  7  hundredths  are  49 
hundredths,  and  3  (which  I  reserved)  are  52  hundredths, 
that  is,  5  tenths  and  2  hundredths.  I  write  the  two  hun- 
dredths, reserving  the  5  tenths.  Then  7  times  8  tenths  are 
56  tenths,  and  5  (which  I  reserved)  are  61  tenths,  that  is,  6 
whole  ones  and  1  tenth.  I  write  the  1  tenth,  reserving  the 
6  units.  Then  7  times  9  are  63,  and  6  are  69,  &c.  It  is 
evident  then,  that  there  must  be  thousandths  in  the  product, 
as  there  are  in  the  multiplicand.  The  point  must  be  made 
between  the  third  and  fourth  figure  from  the  right,  as  in 
the  multiplicand,  and  the  answer  will  stand  thus,  139.125 
yards. 

Rule.  When  there  are  decimal  figures  in  the  multipli- 
cand only,  cut  off  as  many  places  from  the  right  of  the  pro- 
duct for  decimals,  as  there  are  in  the  multiplicand. 

If  a  ship  is  worth  24683  dollars,  what  is  a  man's  share 
worth,  who  own'^  |  of  her. 

J  =  .375  =:  xV/o-     The  question  then  is,  to  find  ^V^  of 


XXVII.  DECIMALS.  197 

246S3  dollars.  First  find  j-^'^  of  it,  that  is,  divide  it  by  1000. 
This  is  done  by  cutting  off  tliree  places  from  the  riirht  (Art. 
XI.)  thus  •^4.(>8;3,  that  is,  2i^%%,  because  68:J  is  a  remainder 
and  must  be  written  over  the  divisor.  In  fact  it  h  evident 
that  J Jou  of  24083  is  \*,%\'  =  24^VoV  ^^^  since  this  frac- 
tion is  thousandths,  it  may  stand  in  the  form  of  a  decimal, 
thus  24.683. 

It  is  a  general  rule  then,  that  when  wc  divide  hy  10.  100, 
1000,  t^c.  which  is  done  hy  cutting  off  Jigures  from  the 
right,  the  figures  so  cut  off  may  stand  as  decimals,  because 
they  ivill  always  be  tenths,  hundredths,  S^c. 

^^Vu  of  24083  then  is  24.683  and  -j-^/^  of  it  will  be  375 
times  24.683.  Therefore  24.683  must  be  multiplied  by 
375, 

24.683  24683 

375  .375 


123415  123415 

172781  172781 

74049  74049 


$9256.125  Ans.  $9256.125 

This  result  must  have  three  decimal  places,  because  the 
multiplicand  has  three.  The  answer  is  9256  dollars,  12 
cents,  and  5  mills.  But  the  purpose  was  to  multiply  24683 
by  .375,  in  which  case  the  multiplier  has  three  decimal 
places,  and  the  multiplicand  none.  We  pointed  off  as  many 
places  from  the  right  of  the  multiplicand,  as  there  were  in 
the  multiplier,  and  then  used  the  multiplier  as  a  whole  num- 
ber. This  in  fact  makes  the  same  number  of  decimal 
places  in  the  product  as  there  are  in  the  multiplier. 

We  may  arrive  at  this  result  by  another  mode  of  reason- 
ing. Units  multiplied  by  tenths  will  produce  tenths  ;  units 
multiplied  by  hundredths  will  produce  hundredths ;  units 
multiplied  by  thousandths  will  produce  thousandths,  &c. 

In  the  second  operation  of  the  above  example,  observe, 
that  .375  is  j\,  and  ^^^,  and  ^\^,  then  -^^\^  of  3  is  y/^^, 
and  j-^-^-^^  of  3  is  y^^o,  which  is  -, ^-^  and  yoVo>  ^^^  down  the 
5  thousandths  in  the  place  of  thousandths,  reserving  the  yi^; 
Then  ^^^  of  80  is  yl^^,  or  -,1-^,  and  5  times  ^f^  is  i\\, 
and  yi^i  (which  was  reserved)  are  y^J^-,  equal  to  >„  ^"^  yio- 
Set  down  the  ^wo  '"  ^^^  hundredth's  place,  &c.  This  shows 
also,  that  lohen  there  are  no  decimals  in  the  multiplicands 
17  * 


198  ARITHMETIC.  Part^ 

there  must  he  as  many  decimal  places  in  the  product  as  in  tht 
multiplier. 

It  was  observed  that  when  a  whole  number  is  to  be  multi- 
plied by  10,  100,  &c.  it  is  done  by  annexing  as  many  zeroH 
to  the  right  of  the  number  as  there  are  in  the  multiplier,  and 
to  divide  by  these  numbers,  it  is  done  by  cutting  off  as  many 
places  as  there  are  zeros  in  the  divisor.  When  a  number 
containing  decimals  is  to  be  multiplied  or  divided  by  10,  100, 
&,c.  it  is  done  by  removing  the  decimal  point  as  many  placets 
to  the  right  for  multiplication,  and  to  the  left  for  division,  as 
there  are  zeros  in  the  multiplier  or  divisor.  If,  for  example, 
we  wish  to  multiply  384.785  by  10,  we  remove  the  point  one 
place  to  the  right,  thus,  3847.85,  if  by  100,  we  remove  it 
two  places,  thus,  38478.5.  If  we  wish  to  divide  the  same 
number  by  10,  we  remove  the  point  one  place  to  the  left, 
thus,  38.4785  ;  if  by  100,  we  remove  it  two  places,  thus, 
3.84785.  The  reason  is  evident,  for  removing  the  point 
one  place  towards  the  right,  units  become  tens,  and  the 
the  tenths  become  units,  and  each  figure  in  the  number  is 
increased  tenfold,  and  when  removed  the  other  way  each 
figure  is  diminished  tenfold,  &c. 

How  much  cotton  is  there  in  2^^  hales,  each  hale  contain- 
ing 4|  twt, 

3/^z=3.7;  4f  =  4.75, 
In  this  example  there  are  decimals  in  both  multiplicand 
and  multiplier. 

4,75 
3.7 

3325 
1425 


Ans.  17.575  cwt. 

3.7  13  the  same  as  |J,  we  have  to  find  U  of  4.75.  Now 
^\)  of  4.75,  we  have  just  seen,  must  be  .475,  and  f  J  is  37 
limes  as  much.  We  must  therefore  multiply  .475  by  37, 
which  gives  17.575  cwt. 

We  shall  obtain  the  same  result  if  we  express  the  whole 
in  the  form  of  common  fractions.  4.75  =:  4/^  =  f  ^,  and 
3.7  =  f  ^.  Now  according  to  Art.  XVII.  ^V  of  f^f  is  j'^%, 
and  ^  will  be  37  limes  as  much,  that  is  VoW  =  ^'^t/o% 
=  17.575  as  before. 


XXVII.  DECIMALS.  199 

In  looking  over  the  above  process  we  find,  that  the  two 
numbers  are  multiplied  together  in  the  saine  manner  as  whole 
numbers,  and  as  mani/  places  are  pointed  off  for  decimals  in 
the  product,  as  there  are  in  the  multiplicand  and  multiplier 
counted  together. 

It  is  plain  that  this  must  always  be  the  case,  for  tenths 
multiplied  by  tenths  must  produce  tenths  of  tenths,  that  is 
hundredths,  which  is  two  places  ;  tenths  multiplied  by  hun- 
dredths must  produce  tenths  of  hundredths,  or  thousandths, 
which  is  three  places  ;  hundredths  multiplied  by  hundredths 
must  produce  hundredths  of  hundredths,  that  is  ten-thou- 
sandths, which  is  four  places,  &c. 

What  cost  5|  tons  of  hay,  at  $27.38  per  ton  ?  5f  = 
5.375. 

27.38 
5.375 


13690 
19166 
8214 

13690 


$147.16750  Ans. 

In  this  example  there  are  hundredths  in  the  multiplicand, 
and  thousandths  in  the  multiplier.  Now  hundredths  multi- 
plied by  thousandths  must  produce  hundredths  of  thou- 
sandths, winch  is  five  decimal  places,  the  number  found  by 
counting  the  places  in  the  multiplicand  and  multiplier  to- 
gether. The  answer  is  147  dollars,  16  cents,  7  mills,  and 
3^  of  a  mill. 

A  man  owned  .03  of  the  stock  in  a  bank,  and  sold  .2  of 
kis  share.     What  part  of  the  whole  stock  did  he  sell  ? 

It  is  evident  that  the  answer  to  this  question  must  be  ex- 
pressed in  thousandths,  for  hundredths  multiplied  by  tenths 
must  produce  thousandths.  -^^  of  y§^  are  yoW-  ^"*  *^  w® 
multiply  them  in  the  form  of  decimals,  we  obtain  only  one 
figure,  viz.  6.  In  order  to  make  it  express  -3-o*Vo  *'  will  be 
necessary  to  write  two  zeros  before  it,  thus,  .006. 

.03 

.2 

Ans.  .006  of  the  whole  stock. 


200  ARITHMETIC.  Part  2. 

This  result  is  agreeable  to  the  above  rule. 

The  following  is  the  general  rule  for  multiplication,  when 
there  are  decimals  in  either  or  both  the  numbers  :  Multiply 
as  in  whole  nmnhcr^,  and  point  off  as  many  places  from  the 
right  of  the  product  for  decimals^  as  there  are  decimal 
places  in  the  multiplicand  and  multiplier  counted  together. 
If  the  product  does  not  contain  so  many  places,  as  many 
zeros  must  he  written  at  the  left,  as  are  necessary  to  make  up 
the  number. 


Division  of  Decimals. 

XXVIII.      A   man  bought   8   yards   of  broadcloth  for 
$75.376 ;  how  much  was  it  per  yard  7 

$75,376 
mills.  75376  (8 

72         

9422  mills. 

33 

32  $9,422  Ans. 

17 
16 

16 
16 


In  this  example  there  are  decimals  in  the  dividend  only. 

I  consider  $75,376  as  75376  mills.  Then  dividing  by  8, 
either  by  long  or  short  division,  I  obtain  9422  mills  pet- 
yard,  which  is  $9,422.  The  answer  has  the  same  number 
of  decimal  places  as  the  dividend. 

Divide  117.54  bushels  of  corn  equally  among  18  men. 
How  much  loill  each  have  1 

117.54  =  117/o\  =  HH*  ;  this  divided  by  18  gives 
K-^=6/oV^6.53. 


XXVIII.  DECIMALS.  201 

117.54  (18 
108       


6.53 


95 
90 

54 
54 


Or  we  may  reason  as  follows.  I  diviJe  117  by  18,  which 
gives  6,  and  9  remainder.  9  whole  ones  are  90  tenths,  and 
5  are  95  tenths  ;  this  divided  by  IS  gives  5,  which  must  be 
tenths,  and  5  remainder.  5  tenths  are  50  hundredths,  and 
4  are  54  hundredths;  this  divided  by  18  gives  3,  which 
must  be  3  hundredths.     The  answer  is  6.53  each,  as  before. 

If  you  divide  7.75  barrels  of  jlour  equally  among  13  wic», 
Iwio  much  will  you  give  each  of  them  7 

7.75  (13 
65     


.596  + 

125 
117 

80 

78 


It  is  evident  that  they  cannot  have  t30  much  as  a  barrel 
each.  7.75  =.  l^l  =  i^f  ^  Dividing  this  by  13,  I  obtain 
j5_9_6^  and  a  small  lemainder,  which  is  not  worth  noticing, 
since  it  is  only  a  part  of  a  thousandth  of  a  barrel.  -^^^-^  = 
.596.  Or  we  may  reason  thus  :  7  whole  ones  are  70  tenths, 
and  7  are  77  tenths.  This  divided  by  13  gives  5,  which 
must  be  tenths,  and  12  remainder.  12  tentlis  are  120  hun- 
dredths, and  5  are  125  hundredths.  This  divided  by  13 
gives  9,  which  must  be  hundredths,  and  8  remainder.  We 
may  now  reduce  this  to  thousandths,  by  annexing  a  zero,  8 
hundredths  are  80  thousandths.  This  divided  by  13  gives  6, 
which  must  be  thousandths,  and  2  remainder.  Thousandths 
will  be  sufficiently  exact  in  this  instance,  we  may  therefore 


am  ARITHMETIC.  Pari  2. 

omit  the  remainder.     The  answer  is  .596  -f-  <^f  a  barrel 
each. 

From  the  above  examples  it  appears,  that  when  only  the 
dividend  contains  dccima/s,  division  is  performed  as  in  loholc 
numbers^  mid  in  the  result  as  many  decimal  places  must  he 
pointed  off  from  the  right,  as  there  are  in  the  dividend. 

Note.  If  there  be  a  remainder  after  all  the  figures  have 
been  brought  down,  the  division  may  be  carried  further,  by 
annexing  zeros.  In  estimating  the  decimal  places  in  the 
quotient,  the  zeros  must  be  counted  with  the  decimal  places 
of  the  dividend. 

At  $6.75  a  cord,  how  many  cords  of  7cood  may  he  bought 

for  $;^8  ? 

In  this  example  there  are  decimals  in  the  divisor  only. 
$6.75  is  675  cents  or  \^  of  a  dollar.  The  88  dollars  must 
also  be  reduced  to  cents  or  hundredths.  This  is  done  by 
annexing  two  zeros.  Then  as  many  times  as  675  hun- 
dredths are  contained  in  3800  hundredths,  so  many  cords 
may  be  bought. 

3800  (675         or         3800  (675 

3375    3375  

5|-f  I  cords.   5.62  -j-  cords. 

4-25  4250 

4050 


2000 
1350 

650 


The  answer  is  5ff  |^  cords,  or  reducing  the  fraction  to  a 
decimal,  by  annexing  zeros  and  continuing  the  division, 
5.62  -f  cords. 

If  3.423  yards  of  cloth  cost  $25,  what  is  iliat  per  yard  ? 

3.423  =  3A\V  =  4m- 

The  question  is,  if  ft|J  of  a  yard  cost  $25,  what  is  that  a 
yard  1 

According  to  Art.  XXIV.,  we  must  multiply  25  by  1000, 
that  is,  annex  three  zeros,  and  divide  by  3423. 


^IIl.               ] 

25000  (3423 
23961  

or 

[ALS. 

25000  (3423 

23961  

7  30  !  AnR_ 

1039 

10390 
10269 

203 


121 

The  answer  is  $7i^|,  or  reducing  the  fraction  to  cents, 
$7.30  per  yard. 

If  1.875  yard  of  cloth  is  sufficient  to  make  a  coat ;  how 
many  coats  may  be  made  of  47.5  yards  1 

In  this  example  the  divisor  is  thousandths,  and  the  divi- 
dend tenths.  If  two  zeros  be  annexed  to  the  dividend  it 
will  be  reduced  to  thousandths. 

47.500  (1.875  or  47500  (1875 

3750      3750      

25,W5 25.33  + 

10000  10000 

9375  9375 


625  6250 

5625 


6250 
5625 

625 


1875  thousandths  are  contained  in  47500  thousandths 
25  ^^Vj  times,  or  reducing  the  fraction  to  decimals,  25.33  -f- 
times,  consequently,  25  coats,  and  y/^  of  another  coat  may 
be  made  from  it. 

From  the  three  last  examples  we  derive  the  following  rule  : 
When  the  divisor  only  contains  decimals,  or  ichen  there  are 
more  decimal  places  in  the  divisor  than  in  the  dividend,  an- 
nex as  many  zeros  to  the  dividend  as  the  places  in  the  divisor 
exceed  those  in  the  dividend,  and  then  proceed  as  in  whole 
numbers.      The  answer  will  be  whole  numbers. 

At  $2.25  per  gallon,  hoio  many  gallons  of  wine  may  he 
bought  for  $15,375  ? 


«04  ARITHMETIC.  Part  2. 

In  this  example  the  purpose  is  to  find  how  many  times 
$2.25  is  contained  in  $15,375.  There  are  more  decimal 
places  in  the  dividend  than  in  the  divisor.  The  first  thing 
that  suggests  itself,  is  to  reduce  the  divisor  to  the  same  de- 
nomination as  the  dividend,  that  is,  to  mills  or  thousandths. 
This  is  done  by  annexing  a  zero,  thus,  $2,250.  The  ques- 
tion  is  now,  to  find  how  many  times  2250  mills  are  contain- 
ed in  15375  mills.  It  is  not  important  whether  the  point 
be  taken  away  or  not. 

15375  (2250 

13500   

6.83  +  gals.    Ans. 


18750 
18000 


7500 
6750 

750 

Instead  of  reducing  the  divisor  to  mills  or  thousandths, 
we  may  reduce  the  dividend  to  cents  or  hundredths,  thus, 
$15,375  are  1537.5  cents.  The  question  is  now,  to  find 
how  many  times  225  cents  are  contained  in  1537.5  cents. 
This  is  now  the  same  as  the  case  where  there  were  deci- 
mals in  the  dividend  only,  the  divisor  being  a  whole  num- 
ber. 

1537.5  (225 
1350      

6.83  -f-  gals.     Ans.  as  before. 

1875 

1800 

750 
675 

75 

Jf  3.15  bushels  of  oats  will  keep  a  horse  1  week,  how  many 
weeks  will  37.5764  bushels  keep  him  ? 

The  question  is,  to  find  how  many  times  3.15  is  contained 
in  37.5764.  The  dividend  contains  ten  thousandths.  The 
divisor  is  31500  ten  thousandths. 


XX  VIII.  DECIMALS.  5805 

375764  (31500 
31500     


60764 
31500 

292640 
283500 


11.929  + weeks.  Ans. 


91400 
63000 

284000 
283500 

500 

Instead  of  reducing  the  'iivisor  to  ten-thousandths,  we  may 
reduce  the  dividend  to  hundredths.  37.5764  are  3757.64 
hundredths  of  a  bushel.  The  decimal  .64  in  this,  is  a  frac- 
tion of  an  hundredth. 

3.15  are  315  hundredths.  Now  the  question  is,  to  find 
how  many  times  315  hundredths  are  contained  in  3757.64 
hundredths.  ^ 

3757.64  (315 
315  


607 
315 

2926 
2835 


11.929  +  weeks.     Ans.  as  before. 


914 
630 

2840 
2835 


From  the  two  last  examples  we  derive  the  following  ni?c 
for  division  :   When  the  dividend  contains  more  decimal  places 
18 


206  ARITHMETIC.  Pari  2. 

than  the  divisor :    Reduce  them  both  to  the  same  denomina- 
tion, and  divide  as  in  ichole  numhers. 

IS.  B.  There  are  two  ways  of  reducing  them  to  the  same 
denomination.  First,  the  divisor  may  be  reduced  to  the 
same  denomination  as  the  dividend,  by  annexing  zeros,  and 
taking  away  the  points  from  both.  Secondly,  the  dividend 
may  be  reduced  to  the  same  denomination  as  the  divisor,  by 
taking  away  the  point  from  the  divisor,  and  removing  it  in 
the  dividend  towards  the  right  as  many  places  as  there  are 
in  the  divisor.     The  second  method  is  preferable. 

The  same  resuit  may  be  produced  by  another  mode  of 
reasoning.  The  quotient  must  be  such  a  number,  that  be- 
ing multiplied  with  the  divisor  will  reproduce  the  dividend 
Now  a  product  must  have  as  many  decimal  places  as  there 
are  in  the  multiplier  and  multiplicand  both.  Consequently 
the  decimal  places  in  the  divisor  and  quotient  together  must 
be  equal  to  those  m  the  dividend.  In  the  last  example  there 
were  four  decimal  places  in  the  dividend  and  two  in  the  di- 
visor ;  this  would  give  two  places  in  the  quotient.  Then  a 
zero  was  annexed  in  the  course  of  the  division,  which  made 
three  places  in  the  quotient.  The  rule  may  be  expressed  as 
follows : 

Divide  as  in  whole  numbers,  and  in  the  residi  point  off  as 
many  places  for  decimals  as  those  in  the  dividend  exceed 
those  in  the  divisor.  If  zeros  are  annexed  to  the  dividend, 
count  them  as  so  many  decimals  in  the  dividend.  If  there  are 
not  so  many  places  in  the  result  cts  crre  required^  they  must  be 
supplied  by  writing  zeros  on  the  left. 

Division  in  decimals,  as  well  as  in  whole  numbers,  may 
be  expressed  in  the  form  of  common  fractions. 

What  part  of  .5  is  .3  ?     Ans.  |. 
What  part  of  .08  is  .05  ?     Ans.  f . 
What  part  of  .19  is  .43  ?     Ans.  f|. 
What  part  of  .3  is  .07? 

To  answer  tliis,  .3  must  be  reduced  to  hundredths.     .3  ib 
.30,  the  answer  therefore  is  y^* 
What  part  of  14.035  is  3.S? 
3.8  is  3.800,  the  answer  therefore  is  fVorV 

In  fine,  to  express  the  division  of  one  number  by  another, 
when  either  or  both  contain  decimals,  reduce  them  both  to  the 


XXIX.  DECIMALS.  207 

lowest  denomination  mentioned  in  either,  and  then  write  the 
divisor  under  the  dividend,  as  if  they  were  whole  numbers. 


Circulating  Decimals* 

XXIX.  There  are  some  common  fractions  which  cannot 
be  expressed  exactly  in  decimals.  If  we  attempt  to  change 
\  to  decimals  for  example,  we  find  .3333,  &.c.  there  is  always 
a  remainder  1,  and  the  same  figure  3  will  always  be  repeated 
however  far  we  may  continue  it.  At  each  division  we  ap- 
proximate ten  times  nearer  to  the  true  value,  and  yet  we  can 
never  obtain  it.  \  z=  .1666,  &/C. ;  this  begins  to  repeat  at 
the  second  figure,  y^y  =:  .545454,  &c. ;  this  repeats  two 
figures.  In  the  division  the  remainders  are  alternately  6 
and  5.  3^3  =z  .168163,  &C.  ;  this  repeats  three  figures, 
and  the  remainders  are  alternately  56,  227,  and  272.  Some 
do  not  begin  to  repeat  until  after  two  or  three  or  more 
places.  It  is  evident  that  whenever  the  same  remainder  re- 
curs a  second  time,  the  quotient  figures  and  the  same  remain- 
ders will  repeat  over  again  in  the  same  order.  In  the  last 
example  for  instance,  the  number  with  which  we  commenc- 
ed was  56 ;  we  annexed  a  zero  and  divided  ;  this  gave  a 
quotient  1,  and  a  remainder  227  ;  we  annexed  another  zero, 
and  the  quotient  was  6,  and  the  remainder  272  ;  we  annex- 
ed another  zero,  and  the  -quotient  was  8,  and  the  remainder 
56,  the  number  we  commenced  with.  If  we  annex  a  zero 
to  this,  it  is  evident  that  we  shall  obtain  the  same  quotient 
and  the  same  remainder  as  at  first,  and  that  it  will  continue 
to  repeat  the  same  three  figures  for  ever. 

It  is  evident  that  the  number  of  these  remainders,  and 
consequently  the  number  of  figures  which  repeat,  must  be 
one  less  than  the  number  of  units  in  the  divisor.  If  the 
fraction  is  4,  there  can  be  only  six  different  remainders ; 
after  this  number,  one  of  them  must  necessarily  recur  again, 
and  then  the  figures  will  be  repeated  again  in  the  same  or 
der. 


208  ARITHMETIC.  Part  2 

1     (7 
10  — 
7  .1428571,  &c. 

—  It  commences  with  1  for  the 
30  dividend,  then  annexing  zeros, 
28                            the  remainders  are  3,  2,  6,  4,  5, 

which  are  all  the  numbers  below 

20  7 ;  then  comes  1  again,  the  num- 
14  ber  with  which  it  commenced, 
and  it  is  evident  the  whole  will  be 

60  repeated  again  in  the  same  order. 
56  Decimals  which  repeat  in  this 
way  are   called  circulating  deci. 

40  mcds, 

35 

50 
49 

10 

7 

3 

Whenever  we  find  that  a  fraction  begins  to  repeat,  we 
may  write  out  as  many  places  as  we  wish  to  retain,  without 
the  trouble  of  dividing. 

As  it  is  impossible  to  express  the  value  of  such  a  fraction 
by  a  decimal  exactly,  rules  have  been  invented  by  which 
operations  may  be  performed  on  them,  with  nearly  as  much 
accuracy  as  if  they  could  be  expressed  ;  but  as  they  are  long 
and  tedious,  and  seldom  used,  I  shall  not  notice  them.  Suf- 
ficient accuracy  may  always  be  attained  without  them. 

I  shall  show,  however,  how  the  true  value  of  them  may 
always  be  found  in  common  fractions. 

The  fraction  ^  reduced  to  a  decimal,  is  .1111  .  .  .  &c. 
Therefore,  if  we  wish  to  change  this  fraction  to  a  common 
fraction,  instead  of  calling  it  j\,  y^V,  or  ^U_,  which  will  be 
a  value  too  small,  whatever  number  of  figures  we  take,  we 
must  call  it  -i-.  This  is  exact,  because  it  is  the  fraction 
which  produces  the  decimal.  If  we  have  the  fraction  .2222 . . 
&c.  It  is  plain  that  this  is  twice  as  much  the  other,  and  must 
be  called  f .     If  |  be  reduced  to  a  decimal,  it  produces  .2222 

.  &c.     If  we  have  .3333  .  .  &,c.  this  being  tliree  times  ai 


XXIX.  DECIMALS.  209 

much  as  the  first,  is  |  rz:  i.  If  i  be  reduced  to  a  decimal,  it 
produces  .3333  .  .  &c.  It  is  plain,  that  whenever  a  single 
figure  repeats,  it  is  so  many  ninths. 

Change  .4444  &lc.  to  a  common  fraction.     Ans.  ^. 

Change  .5555  &lc.  to  a  common  fraction. 

Change  .6666  &lc.  to  a  common  fraction. 

Change  .7777  &c.  to  a  common  fraction. 

Change  .9999  &lc.  to  a  common  fraction. 

Change  .5333  d:c.  to  a  comiv^on  fraction. 

This  begins  to  repeat  at  the  second  figure  or  hundredths^ 
The  first  figure  5  is  -^^ ;  and  the  remaining  part  of  the  frac- 
tion is  I  of  y\,  that  is,  -^  =  -^^-^ ;  these  must  be  added  to- 
gether, /o-  is  if,  and  3'^  makes  4-^-  =  1%.  The  answer  is 
^\.  If  this  be  changed  to  a  decimal,  it  will  be  found  to  be 
.5333  &c. 

If  a  decimal  begins  to  repeat  at  the  third  place,  the  two 
first  figures  will  be  so  many  hundredths,  antl  the  repeating 
figure  will  be  so  many  ninths  of  another  hundredth. 

Change  .4660  &,c.  to  a  common  fraction. 

Change  ,3888  &c.  to  a  common  fraction. 

Change  ..3744  &lc.  to  a  common  fraction. 

Change  .46355  &c.  to  a  common  fraction. 

If  ^'^  be  changed  to  a  decimal,  it  produces  .010101  &c. 
The  decimal  .030303  &.c.  is  three  times  as  much,  therefore 
It  must  be  -^  =  ^\.  The  decimal  .363636  &.c.  is  thirty-six 
times  as  much,  therefore  it  nmst  be  y|  ^=  tt* 

If  g-ro  he  changed  to  a  decimal,  it  produces  .001001001 
&c.  The  decimal  .006006  &,c.  is  6  times  as  much,  there- 
fore it  must  be  -^  zzz  ^f  ^.  The  fraction  .027027  &-c.  is 
twenty-seven  times  as  much,  and  must  be  -^^  =:  y|y.  The 
fraction  .354354  &.c.  is  3.54  times  as  much,  and  must  be 
3X|.  zr:  i||.  This  principle  is  true  for  any  number  of  places. 
Hence  we  derive  the  following  rule  for  changing  a  circulat- 
ing decimal  to  a  common  fraction  :  Make  the  repeating 
figures  the  numerator^  and  the,  denominator  ivill  be  as  mamy 
9s  as  there  are  repeating  figures. 

If  they  do  not  l)egin  to  repeat  at  the  first  place,  the  pre- 
ceding figures  must  be  called  so  many  tenths,  hundredths,  S^c, 
according  to  their  number,  then  the  repeating  part  must  be 
changed  in  the  above  ynanner,  but  instead  of  being  the  frac- 
tion of  an  unit,  rt  loill  be  the  fraction  of  a  tenth,  hundredth^ 
Sfc.  according  to  the  place  in  which  it  commences. 

Instead  of  writing  the  repeating  figures  over  several  times, 
18  * 


21 0  ARITHMETIC.  Part  2. 

they  are  sometimes  written  with  a  point  over  the  first  and 
last  to  show  which  figures  repeat.  Thus  .333  &,c.  is  writ- 
ten .3.  .2525  &LC,  is  written  .25*.  .387387  &c.  is  written 
.387.     .57346340  &c.  is  written  .57346. 

Change  .24  to  a  common  fraction. 

Change  .42  to  a  common  fraction. 

Change  .537  to  a  common  fraction. 

Change  .4745  to  a  common  fraction. 

Change  .8374  to  a  common  fraction. 

Change  .47647  to  a  common  fraction. 

Note.  To  know  whether  you  have  found  the  right  an- 
swer, change  the  common  fraction,  which  you  have  found,  to 
a  decimal  again.     If  it  produces  the  same,  it  is  right. 

Proof  of  Multiplication  and  Division  hy  casting  out  9s. 

If  either  the  muUiplicand  or  the  multiplier  be  divisible  by 
9,  it  is  evident  the  product  must  be  so. 
Multiply  437  by  85. 

437  81  times  437  —  35397 

85  4  times  432=    1728 

4  times      5  =        20 

2185  


3496  37145 


Ans.  37145 

85  =  81  +  4,  and  437  =:  432  +  5.  81  is  divisible  by  9, 
and  85  being  divided  by  9  leaves  a  remainder  4.  432  is  di- 
visible by  9,  and  437  leaves  a  remainder  5.  81  times  437, 
and  4  times  432,  and  4  times  5,  added  together,  are  equal  to 
85  times  437.  81  times  437  is  divisible  by  9,  because  81  is 
so,  and  4  times  432  is  divisible  by  9,  because  432  is  so.  The 
only  part  of  the  product  which  is  not  divisible  by  9,  is  the 
product  of  the  two  remainders  4  and  5.  This  product,  20, 
divided  by  9,  leaves  a  remainder  2.  It  is  plain,  therefore, 
that  if  the  whole  product,  37145,  be  divided  by  9,  the  re- 
mainder must  be  2,  the  same  as  that  of  the  product  of  the 
remainder. 

Therefore  to  prove  multiplication,  divide  the  divisor  and 
the  dividend  hy  9^  and  multiply  the  remainders  together^  and 


Part  2.  ARITHMETIC.  211 

divide  the  product  by  9,  and  note  the  remainder ;  then  divide 
the  lohole  product  by  0,  and  if  the  remainder  is  the  same  as 
the  last,  the  roork  is  right. 

Instead  of  dividing-  by  9,  the  figures  of  each  number  may 
be  added,  and  their  sum  be  divided  by  9,  as  in  Art.  XXI., 
(and  for  the  same  reason)  and  the  remainders  will  be  the 
same  as  if  the  numbers  themselves  were  divided. 

In  the  above  example,  say  7  and  3  and  4  are  14,  which,, 
divided  by  9,  leaves  a  remainder  5 ;  then  5  and  8  are  13, 
which,  divided  by  9,  leaves  a  remainder  4.  Then  4  times 
5  are  20,  which,  divided  by  9,  leaves  a  remainder  2.  Then 
adding  the  figures  of  the  product,  5  and  4  and  1  and  7  and 
3  are  20,  which  being  divided  by  9  leaves  2,  as  the  other. 
Instead  of  dividing  14  and  13  by  9,  these  figures  may  be 
added  together,  thus  4  and  1  are  5  ;  3  and  1  are  4. 

Since  in  division  the  quotient  multiplied  by  the  divisor 
produces  the  dividend  ;  if  the  divisor  and  quotient  be  divided 
by  9  and  the  remainders  multiplied  together,  and  this  pro- 
duct divided  by  9,  and  the  remainder  noted;  and  then  the 
dividend  be  divided  by  9  ;  this  last  remainder  must  agree  with 
the  other, 

N.  B.  If  there  is  a  remainder  after  division,  it  must  be 
subtracted  from  the  dividend  before  proving  it. 


Miscellaneous  Examples. 

1.  If  2  lbs.  of  figs  cost  2s.  8d.,  what  is  that  per  lb.  1 

2.  If  2  bushels  of  corn  cost  8s.  6d.,  what  is  that  pel* 
bushel  \ 

3.  If  2  lbs.  of  raisins  cost  Is.  lOd.,  what  is  that  per  lb.  1 

4.  If  3  bushels  of  potatoes  cost  9s.  6d.,  what  is  that  per 
bushel  ? 

5.  If  4  gals,  of  gin  cost  12s.  8d.,  what  is  that  per  gal.  1 

6.  If  2  barrels  of  flour  cost  3:^.  4s.,  what  is  that  per  bar- 
rel? 

7.  If  2  gallons  of  wine  cost  \£.  10s.  4<i.,  what  is  that  per 
gallon  1 

8.  If  2  barrels  of  beer  cost  \£.  15s.  8d.,  what  is  that  per 
barrel  ? 

9.  If  4  gallons  of  gin  cost  17s.' 8d.,  what  is  that  per  gallon  \ 


212  ARITPIMETIC.  Part  2. 

10.  Ii  D  yards  of  cloth  cost  6^.  10s.  5d.,  what  is  that  per 
yard  1 

11.  If  7  barrels  of  flour  cost  V7£.  Ss.  7d.,  what  is  that  per 
barrel  ? 

12.  If  8  yards  of  cloth  cost  20c£.  18s.  5.,  what  is  that  per 
yard  ? 

'  13.  A  man  had  4  cwt.  3  qrs.  14  lbs.  of  tobacco,  which  he 
put  into  2  boxes,  ^  of  it  in  each  ;  how  much  did  he  put  in 
each  box  1 

14.  Divide  VZ£.  8s.  5d.  equally  among  5  men. 

15.  Divide  8  cwt.  3  qrs.  17  lbs.  into  3  equal  parts. 

16.  Divide  Ifi  cwt.  1  qr.  11  lbs.  of  flour  equally  among  7 
men  ;  how  much  will  each  have  ? 

17.  Divide  3  hhds.  42  gals.  2  qts.  into  5  equal  parts. 

18.  If  12  yards,  3  qrs.  2  nls.  of  cloth  will  make  7  coats, 
how  much  will  make  1  coat  1  How  much  will  make  13 
coats  1 

19.  If  5  yards  of  cloth  cost  19^.  3s.  4d.,  what  cost  17 
yards  1 

20.  What  is  I  of  45^.  9s.  7d.  ? 

21.  If  18  cwt.  of  sugar  cost  56<£.  13s.  8d.  what  will  53| 
cwt.  cost  1 

22.  If  ^  of  a  ship  is  worth  943^.  7s.  8d.,  what  is  the  whole 
ship  worth  ? 

23.  If  84  cows  cost  453.£.  14s.  8d.,  how  much  is  that 
apiece  ? 

24.  If  3^  cwt.  of  sugar  cost  9^.  15s.  9d.,  what  is  that  per 
cwt.  ? 

25.  If  9|-  barrels  of  flour  cost  21=£.  3s.  8d.,  what  cost  17| 
barrels  ? 

26.  If  a  staflT  4  feet  long  cast  a  shade  on  level  ground  6 
ft.  8  in.,  what  is  the  height  of  a  steeple  which  casts  a  shade 
173  feet  at  the  same  time  ? 

27.  If  57  gallons  of  water  in  one  hour  run  into  a  cistern 
containing  258  gallons,  and  by  another  cock  42  gallons  run 
out  in  an  hour,  in  what  time  will  it  be  filled  ? 

23.  A  and  B  depart  from  the  same  place,  and  travel  the 
same  road  ;  but  A  starts  6  days  before  B,  and  travels  at  the 
rate  of  28  miles  a  day  ;  B  follows  at  the  rate  of  43  mdes  a 
day.     In  how  many  days  will  B  overtake  A  1 

29.  A  sets  out  from  Boston  to  New- York,  at  20  min.  past 
8  in  the  morning,  and  travels  at  the  rate  of  5  miles  an  hour; 
and  B  sets  out  from  New- York  to  Boston  at  3  o'clock  in  the 


Vart  %  ARITHMETIC.  213 

afternoon  of  the  same  day,  and  travels  at  the  rate  of  6^  miles 
per  hour.  The  distance  is  250  miles.  Supposing  them  to 
travel  constantly  until  they  meel,  at  what  time  will  they 
meet,  and  at  what  distance  from  each  place  1 

30.  The  distance  from  New- York  to  Baltimore  is  197 
miles.  Two  travellers  set  out  at  the  same  time  in  order  to 
meet ;  A  from  New- York  towards  Baltimore,  and  B  from 
Baltimore  towards  New- York.  When  they  met,  which  was 
at  the  end  of  6  days,  A  had  travelled  3  miles  a  day  more 
than  B.     How  many  miles  did  each  travel  per  day  1 

31.  If  when  wheat  is  7s.  6d.  per  bushel,  the  penny-loaf 
weighs  9  oz.,  what  ought  it  to  weigh  when  wheat  is  6s.  per 
bushel? 

32.  Suppose  650  men  are  in  a  garrison,  and  have  provi- 
sions sufficient  to  last  them  two  months  ;  how  many  men 
must  leave  the  garrison  in  order  to  have  the  provisions  last 
those  who  remain  five  months  ? 

.33.  If  8  boarders  will  drink  a  barrel  of  cider  in  15 
days,  how  long  will  it  last  if  4  more  boarders  come  among 
them? 

34.  A  ship's  crew  of  18  men  is  supposed  to  have  provi- 
sion sufficient  to  last  the  voyage,  if  each  man  is  allowed  23 
oz.  per  day,  when  they  pick  up  a  crew  of  8  persons.  What 
must  then  be  the  daily  allowance  of  each  person  ? 

35.  How  many  yards  of  flannel  that  is  1^  yard  wide  will 
line  a  cloak,  containing  9  yards,  that  is  |  yard  wide  ? 

36.  A  garrison  of  1800  men  have  provisions  sufficient  to 
last,  them  12  months  ;  but  at  the  end  of  3  months,  the  gar- 
rison was  reinforced  by  600  men,  and  2  months  after  that,  a 
second  reinforcement  of  400  men  was  sent  to  the  garrison. 
How  long  did  the  provisions  last  in  the  whole  ? 

37.  A  regiment  of  soldiers,  consisting  of  1000,  are  to  be 
new  clothed ;  each  coat  to  contain  2i  yards  of  cloth  li  yard 
wide,  and  to  be  lined  with  flannel  of  ^  yard  wide.  How 
many  yards  of  flannel  will  line  them  ? 

38.  I  borrowed  185  quarters  of  corn,  when  the  price  was 
19s.  per  quarter  ;  how  much  must  I  pay  to  indemnify  the 
lender  when  the  price  is  17s.  4d.  ? 

39.  If  7  men  can  reap  84  acres  of  wheat  in  12  days,  how 
many  men  can  reap  100  acres  in  5  days  ? 

40.  If  7  men  can  build  36  rods  of  wall  in  3  days,  how 
many  rods  can  20  men  build  in  14  days  ? 

41.  If  20  bushels  of  wheat  are  sufficient  for  a  family  of  15 


214  ARITHMETIC.  Part  % 

persons  3  months,  how  much  will  be  sufficient  for  4  persons 
IJ  months  ? 

42.  If  it  cost  $23.84  to  carry  17  cwt.  3  qrs.  14  lb.  85 
milos,  how  mucli  must  be  paid  for  cariying  53  cwt.  2  qrs.  150 
miles  ? 

43.  If  18  men  can  build  a  wall  40  rods  long,  5  feet  high, 
and  4  feet  thick  in  15  days  ;  in  what  time  will  20  men  build 
one  87  rods  long,  8  feet  high,  snd  5  feet  thick  1 

44.  If  a  family  of  9  persons  spend  $305  in  4  months,  how 
many  dollars  would  maintain  them  8  months,  if  5  persons 
more  were  added  to  the  family  ? 

45.  If  a  regiment  consisting  of  1878  soldiers,  consume 
702  quarters  of  wheat  in  33G  days  ;  how  many  quarters  will 
an  army  of  22536  soldiers  consume  in  112  days  1 

46.  if  12  tailors  can  finish  13  suits  of  clothes  in  7  days, 
how  many  tailors  can  finish  the  clothes  ol  a  regiment  con- 
sisting of  4V)4  soldiers,  in  19  days  of  the  same  length  ? 

47.  If  24  measures  of  wine,  at  3s.  4d.  serve  16  men  for  6 
days,  how  many  measures,  at  2s.  8d.,  will  serve  48  men  4 
days  1 

48  How  many  tiles  8  inches  square,  wiU  cover  a  hearth 
12  feet  wide  and  16  feet  long  ? 

49.  How  many  bricks  9  in.  long,  4i  in.  wide,  and  2  in. 
thick,  will  build  a  wall  6  feet  high  and  13}  in.  thick,  round 
a  garden,  each  side  of  which  is  280  feet  on  the  outside  of  tho 
wall  ? 

50.  There  is  a  house  40  feet  in  length,  and  30  feet  raf- 
ters ;  how  many  shingles  will  it  take  to  cover  the  roof,  sup- 
posing each  shingle  to  be  4  inches  wide,  and  each  course  to 
be  6  inches  1 

51.  A  man  built  a  house  consisting  of  4  stories  ;  in  the 
lower  story  there  were  16  windows,  each  containing  12  panesf 
of  glass,  each  pane  16  in.  long,  12  in.  wide  ;  the  second  and 
third  stories  contg.ined  18  windows,  each  of  the  same  size ; 
the  fourth  sto'-y  contained  18  windows,  each  window  6 
panes  18  by  12.  How  many  square  feet  of  glass  were  there 
in  the  whole  house  ? 

52.  A  merchant  soW  a  piece  of  cloth  for  $40,  and  by  so 
doing  lost  10  per  cent.  He  ought  in  trading  to  have  gained 
15  per  cent.  For  how  much  ought  he  to  have  sold  the 
cloth  ? 

53.  Bought  a  hogshead  of  molasses  for  $25,  but  12  gal- 
lons having  leaked  out,  I  desire  to  sell  the  remainder,  so  as  to 


Part  2.  ARITHMETIC.  215 

gain  3  per  cent,  on  the  whole  cost.     For  how  much  per  gal- 
lon must  I  sell  it  ? 

54.  Bought  a  hogshead  of  brandy,  for  $93  on  6  months' 
credit,  and  sold  it  for  $103  ready  money.  How  much  did  1 
gain,  allowing  money  to  he  worth  O  per  cent,  a  year  ? 

55.  Bought  3  hhds.  of  wine  for  $820  ready  money,  and 
sold  it  at  $1.87  per  gal.  on  6  months'  credit.  What  did  I 
gain,  allowing  money  to  be  worth  6  per  cent,  per  year  ? 

Note.  To  answer  this  question,  it  will  be  necessary  to 
compute  the  interest  on  $320  for  6  months,  and  add  it  to 
$320. 

56.  Bought  a  quantity  of  goods  for  $437.45  and  hired  the 
money  to  pay  for  it,  for  which  I  paid  at  the  rate  of  8  per 
cent,  a  year.  Having  kept  it  on  hand  3  months  and  17 
days,  I  so.  J  it  for  $470,  oo  4  months'  credit.  What  per 
cent,  did  I  gain  1 

57.  Bought  5  hhds.  of  rum  at  1  dollar  per  gal.,  ready 
money,  and  having  kept  it  3  months  and  23  days,  I  sold  it  at 
$1.20  per  gallon,  on  5  months'  credit ;  16  gals,  had  leaked 
out  while  in  my  possession.     How  much  did  I  gain  ? 

When  a  debtor  keeps  money  longer  than  a  year,  the  inter- 
est is  considered  as  due  to  the  creditor  at  the  end  of  the 
year,  and  he  has  a  right  to  demand  it.  If  the  interest  is  not 
paid  at  the  end  of  the  year,  the  creditor  sometimes  requires 
the  interest  for  the  year  to  be  ardded  to  the  principal,  and  con- 
sidered a  part  of  the  debt,  and  consequently  interest  paid 
upon  it  for  the  rest  of  the  time,  and  so  on  at  the  end  of  eve- 
ry year.  In  this  way  the  principal  increases  every  year  by 
the  interest  of  the  last  year.  This  may  seem  just,  but  it  is 
not  allowed  by  law.     This  is  called  compound  interest. 

58.  What  will  $143.17  amount  to  in  3  years  and  4 
months,  at  6  per  cent,  compound  interest  1 

The  most  convenient  method  is,  to  find  the  amount  of  1 
dollar  for  the  time,  and  then  multiply  it  by  the  number  of 
dollars  in  the  question. 


216  ARITHMETIC.  Pert  2. 

1.00 
.06 


.06  interest  for  1  year. 
+  1.00 

=  1.00  amount  for  1  year. 
.06 


.0636  interest  for  2d  year. 
+  1.06 


=  1.1236  amoTiirt  for  2  years. 
.06 


.067416  interest  for  3d  year. 
+  1.1236 

=i  1.191016  amount  for  3  years. 
.02  rate  for  4  months. 


.02382032  interest  for  4  months. 
+  1.191016 


zzz  1.21483632  amount  for  3  years  and  4  months. 
It  will  be  sufficiently  exact  to  use  the  first  four  decimals 
$1.2148.     This  multiplied  by  143.17  will  give  the  answer. 
1.2148 
143.17 


85036 
12148 
36444 
48592 
12148 


$173.922916        Ans.  $173,923—. 

59.  Make  a  table  which  shall  contain  the  amount  of  1  dol- 
lar, for  1  year,  for  two  years,  for  3  years,  &c.  to  20  years, 
at  5  per  cent,  and  at  6  per  cent.  Reserve  fire  decimal 
places. 

N.  B.  The  same  table  will  serve  for  sterling  money,  or 
any  other,  if  the  parts  are  expressed  in  decimals. 


i'art  2. 


ARITHMETIC. 


U\7 


years 

5       rates       6 

years 

5       rates      6 

1 

1.05000 

1.06000 

11 

2 

1.10250 

1.12360 

,12 

3 

13 

' 

4 

i  ^^ 

5 

15 

6 

16 

7 

17 

" 

8 

18 

9 

19 

10 

20 

60.  What  is  the  compound  interest  of  $17.25  for  2  years 
and  7  months,  at  5  per  cent.  ? 

Note.  From  the  table  take  the  amount  of  1  dollar  for 
two  years,  at  5  per  cent,  and  compute  the  interest  on  it  for 
7  months,  at  5  per  cent,  as  in  simple  interest ;  add  this  to 
the  amount  for  two  years.  This  will  be  the  amount  of  1 
dollar  for  2  years  and  7  months.  Multiply  this  by  17.25; 
this  will  be  the  amount  of  $17.25  for  the  time.  Then  to 
find  the  interest,  subtract  the  principal  from  the  amount. 

61.  What  will  $73.42  amount  to  in  4  years,  3  months, 
and  17  days,  at  6  per  cent,  compound*  interest  ? 

62.  A  note  was  given  13th  March,  1815,  for  $847.25 ; 
how  much  had  it  amounted  to  on  the  7th  November,  1820. 
at  6  per  cent,  compound  interest  ? 

63.  How  much  would  the  sum  in  the  last  example  have 
amounted  to  in  the  same  time  at  simple  interest  ? 

64.  What  is  the  compound  interest  of  $1753  for  11  years 
10  months,  and  22  days,  at  6  per  cent.  1 

G5.  A  note  was  given  11th  May,   1813,  for  $847,  rate  6 
per  cent,  compound  interests    The  following  payments  were 
19 


218  ARITHMETIC.  Part  2. 

made;  18th  February,  1815,  $158;  19th  December,  1816 
$87  :  5th  October,  1S19,  $200.  What  was  clue  8th  July, 
1822  ? 

C6.  What  will  \1£.  13s.  fid.  amount  to  in  5  years,  3 
months,  at  6  per  cent,  compound  interest  ? 

Note.  Change  the  shillings  and  pence  to  decimals  of  a 
pound,  nnd  proceed  as  in  Federal  money.  Call  the  unit  in 
the  table  \£.  instead  of  1  dollar. 

67.  What  is  the  compound  interest  of  $643,  for  7  years, 
5  months,  and  18  days,  at  5  per  cent.  ? 

68.  What  is  the  compound  interest  of  143c£.  7s.  4d.  for 
19  years,  7  months,  at  five  per  cent.  1 

69.  A  farmer  mixed  15  bushels  of  rye,  at  64  cents  pei 
bushel  ;  18  bushels  of  corn,  at  55  cents  per  bushel  ;  and  21 
bushels  of  oats,  at  28  cents  per  bushel.  How  many  bushels 
were  there  of  the  mixture  ?  What  was  the  whole  worth  '.' 
What  was  it  worth  per  bushel  T 

70.  A  grocer  mixed  123  lb.  of  sugar,  that  was  worth  8 
cents  per  lb.  ;  87  lb.  that  was  worth  11  cents  per  lb.  ;  and 
15  lb.  that  was  worth  13  cents  per  lb.  What  was  the  mix- 
ture worth  per  lb.  1 

71.  A  grocer  mixed  43  gallons  of  wine,  that  was  worth 
$1.25  per  gal.  with  87  gals,  that  was  worth  $1.60  per  gal. 
What  was  the  mixture  worth  per  gal.  ? 

72.  With  a  hhd.  of  rum,  worth  $.87  per  gal.  a  grocer 
mixed  10  gals,  of  water.  What  was  the  mixture  worth  per 
gal.? 

73.  How  many  gals,  of  rum,  at  $.60  per  gal.  will  come  to 
as  much  as  43  gals,  will  come  to,  at  $.75  per  gal.  ? 

74.  How  much  water  must  be  added  to  a  pipe  of  wine, 
worth  $1.50  per  gal.  in  order  to  reduce  the  price  to  $1.30 
per  gal.  1 

75.  A  grocer  has  two  kinds  of  sugar,  one  at  8  cents  per 
lb.,  the  other  at  13  cents.  He  wishes  to  mix  them  together 
m  such  a  manner,  that  the  mixture  may  be  worth  1 1  cents 
per  lb.  What  will  be  the  proportions  of  each  in  the  mix- 
ture ? 

Note.  The  difference  of  the  two  kinds  is  5  cents.  There- 
fore if  a  pound  of  each  kind  be  divided,  each  into  five  equal 
parts,  the  difference  between  one  part  of  each  will  be  1  cent. 
If -^  lb.  be  taken  from  that  at  8  cents,  and  i  lb.  of  that  at  13 
cents  be  put  in  its  place,  the  pound  will  be  worth  0  cents. 
If  I  lb.  be  taken  from  it,  and  as  much  of  the  other  be  put  in 


Part  2.  ARITHMETIC.  219 

its  place,  the  pound  will  be  worth  1 1  cents,  as  required.  The 
pound  then  will  consist  of  |,  at  8  cents,  and  |,  at  13  cents. 
If  5  lb.  be  mixed,  there  will  be  2  lb.  at  8,"  and  3  at  13 
cents.  The  proportions  are  2  lb.  at  8  as  often  as  3  lb.  at 
13  cents. 

76.  A  farmer  had  oats,  at  38  cents  per  bushel,  which  he 
wished  to  mix  with  corn,  at  75  cents  per  bushel,  so  that  the 
mixture  might  be  50  cents  per  bushel.  What  were  the  pro- 
portions of  the  mixture  1 

Note.  The  difference  in  the  price  of  a  bushel  is  37  cents. 
The  difference  between  -^\  of  a  bushel  of  each  is  1  cent.  If 
■i|-  of  a  bushel  be  taken  from  a  bushel  of  oats,  and  |f  of  a 
bushel  of  corn  be  put  in  its  place,  a  bushel  will  be  formed 
worth  50  cents,  and  consisting  of  if  corn,  and  f-f  oats.  The 
proportions  are  12  of  oats  to  25  of  corn. 

It  is  easy  to  see  that  the  denominator  will  ahcays  he  the 
difference  of  the  prices  of  the  ingredients^  and  the  difference 
between  the  mean  and  the  less  price  will  be  the  numerator 
for  the  quantity  of  the  greater,  and  the  difference  between 
the  mean  and  the  greater  loill  be  the  numerator  for  the  quan- 
tity of  the  less  value.  Take  away  the  denojninators.,  and 
the  numerators  will  express  the  proportions, 

77.  A  merchant  has  spices,  some  at  Od-  per  lb.  some  at 
Is.,  some  at  2s.  and  some  at  2s.  6d.  per  lb.  How  much  of 
each  sort  must  he  mix,  that  he  may  sell  the  mixture  at  la. 
8d.  per  lb.  1 

Note.  Take  one  kind,  the  price  of  which  is  greater,  and 
one,  the  price  of  which  is  less  than  the  mean,  and  find  the 
proportions  as  above.  Then  take  the  other  two  and  find 
their  proportions  in  the  same  way. 


Less  9d.  =  9d. 
Greater  2s.  6d.  =  30d. 


{lid.   diff.  between  less 
and  mean. 
lOd.  diff.  between  great- 
er and  mean. 

The  proportions  are  10  of  the  less  to  11  of  the  greater. 

Less  Is.  =  12d.               1  fSd.    diff.    between    leas 

!  mean  J       and  mean. 

Greater  2s.  =  24d.           [  20      ]  4d.  diff.  between  great- 

\  /      er  and  mean. 


220  ARITHMETIC.  PaH  % 

The  proportions  are  4  of  the  less  to  8  of  the  greater,  which 
is  the  same  as  1  of  the  less  to  2  of  the  greater. 

The  answer  is  10  lb.  at  9d.  to  11  lb.  at  2s.  6d.,  and  1  lb. 
at  Is.  to  2  lb.    at  2s. 

Other  proportions  might  be  found  by  comparing  the  first, 
and  third,  and  the  second  and  fourth. 

78.  A  grocer  has  two  sorts  of  tea,  one  at  75  cents  per  lb. 
and  the  other  at  $1.10  per  lb.  How  must  he  mix  them 
in  order  to  afford  the  mixture  at  $1.00  per  lb.  1 

79.  A  grocer  would  mix  the  following  kinds  of  sugar,  viz. 
at  10  cents,  13  cents,  and  16  cents  per  lb.  What  quantity 
of  each  must  he  take  to  make  a  mixture  worth  12  cents 
per  lb.  ? 

Note.  Those  at  13  and  16  must  both  be  compared  with 
that  at  10  cents  separately. 

80.  A  grocer  has  rum  worth  $.75  per  gal.;  how  many 
parts  water  must  he  put  in,  that  he  may  afford  to  sell  the 
mixture  at  $.65  per  gal.  ? 

81.  It  is  required  to  mix  several  sorts  of  rum,  at  5s.  7d., 
and  9s.  per  gal.  with  water,  so  that  the  mixture  may  be 
worth  6s.  per  gal.  How  much  of  each  sort  musttlie  mixture 
consist  of  ? 

82.  A  farmer  had  10  bushels  of  wheat,  worth  8s.  per 
bushel,  which  he  wished  to  mix  with  corn,  at  3s.  per  bushel, 
so  that  tlie  mixture  might  be  worih  5s.  per  bushel.  How  many 
bushels  of  corn  must  he  use  ? 

Note.  Find  the  proportions  for  a  single  bushel  as  before, 
then  find  how  much  corn  must  be  put  with  1  bushel  of  wheat, 
and  then  with  10  bushels.  The  proportions  are  2  of 
wheat  to  3  of  corn,  consequently  1  of  wheat  to  1^  of  corn, 
and  10  of  wheat  to  15  of  corn; 

83.  A  farmer  would  mix  20  bushels  of  rye,  at  ^  cents  per 
bushel,  with  barley  at  51  cents,  and  oats  at  30  cents  per 
bushel.  How  much  barley  and  oats  must  be  mixed  with 
rye,  that  the  mixture  may  be  worth  41  cents  per  bushel  ? 

84.  A  grocer  had  43. gallons  of  wine  worth  $1.75  per  gal., 
which  he  wished  to  mix  with  another  kind  worth  $1.40  pei- 
gal.,  so  that  the  mixture  might  be  worth  $1.60  per  gal.  How 
many  gals,  of  the  latter  kind  must  he  use  ? 

8.5.  Three  merchants,  A,  B,  and  C,  freight  a  ship  with 
wine.     A  put  on  board  500  tons,  B  340,  and  C  94 ;  in  a 


Part  2.  ARITHMETIC.  "tZX 

storm  they  were  obliged  to  cast  150  tons  overboard.     What 
loss  does  each  sustain  ? 

See    Part  1.      Art.   XVI.,    example    158   and  following. 

86.  A  father  dying,  bequeathed  an  estate  of  $12000  as  fol- 
lows :  \  to  his  wife,  ^  to  his  eldest  son,  \  to  his  second  son, 
and  \  to  his  daughter.  It  is  required  to  divide  the  estate  in 
these  proportions. 

Note.  Reduce  the  fractions  to  a  common  denominator, 
and  the  numerators  will  show  the  proportions. 

87.  Two  men  hired  a  pasture  for  $37,  A  put  in  3  horses 
for  4  months,  and  B  5  horses  for  3  months.  What  ought 
each  to  pay  1 

Note.  3  horses  for  4  months  is  the  same  as  4  times  3  or  12 
horses  for  1  month  ;  and  5  horses  for  3  months,  is  the  same 
as  3  times  5,  or  15  horses  for  1  month.  The  question  there- 
fore is  the  same,  as  if  A  had  put  in  12  horses  and  B  15.  A 
must  pay  \^  and  B  if,  or,  reducing  the  fractions,  f  and  |-. 

88.  Two  men,  A  and  B,  traded  in  company  :  A  put  in 
$350  for  8  months,  and  B  $640  for  5  months ;  they  gained 
$250.     What  was  the  share  of  each  ? 

Note.     Make  the  time  equal,  as  in  the  last  example. 

89.  Four  men  jointly  hired  a  pasture  for  20  English  gui- 
neas ;  A  turned  in  7  oxen  for  13  days,  B  9  oxen  for  14  days, 
C  11  oxen  for  25  days,  and  D  15  oxen  for  37  days.  How 
much  ought  each  to  pay  1 

90.  A  family  of  10  persons  took  a  large  house  for  -^  of  a 
year,  for  which  they  were  to  pay  $500,  for  that  lime.  At  the 
end  of  14  weeks  they  took  in  4  new  lodgers;  and  after  3 
weeks,  4  more  ;  and  so  on  for  every  3  weeks,  during  the 
term,  they  took  in  4  more  lodgers.  What  must  one  of  each 
class  pay  per  week  of  the  rent  ? 

91.  Three  men  enter  into  partnership  and  trade  as  fol- 
lows :  A  put  in  150c£.,  and  at  the  end  of  7  months  took  out 
50c£.  ;  5  months  after  he  put  in  170<£; — B  put  in  20.5i<*., 
and  at  the  end  of  5  months,  llOc£.  more,  but  took  out  \i){)£. 
4  months  after  ; — C  put  in  300  guineas,  at  28s.  each,  and 
wlien  8  months  had  plap-^ed,  he  drew  out  150c£-,  but  5  months 
after  he  put  in  500^*.  Their  partnership  continued  18 
months,  at  the  end  of  which  time  they  had  gained  loO<£. 
Required  each  person's  share  of  the  gain. 


19 


222  ARITHMETIC.  I^art  'Z 

92.  The  last  five  are  examples  of  compound  or  double  feU 
lowship.     What  rule  can  you  make  for  it  ? 

93.  In  how  long  time  will  1  dollar  gain  as  mudi  interest 
AS  $15  ;vill  gain  in  1  month  1 

94.  In  how  long  time  will  1  dollar  gain  as  much  interest 
as  8  dollars  will  gain  in  3  months  1 

95.  In  how  long  time  will  I  dollar  gain  as  much  interest 
as  24  dollars  will  gain  in  5  months  1 

96.  In  how  long  time  will  I  dollar  gain  as  much  interest 
as  $158  will  gain  in  11  months  ? 

97.  In  how  long  time  will  3  dollars  gain  as  much  interest 
as  1  dollar  will  gain  in  24  months  1 

98.  In  how  long  time  will  28  dollars  gain  as  much  interest 
as  1  dollar  will  gain  in  157  months  ? 

99.  A  lent  B  8  dollars  for  2  months,  afterwards  B  lent  A 
1  dollar ;  how  long  ought  he  to  keep  it  to  satisfy  him  for  the 
former  favour  ? 

100.  C  lent  D  1  dollar  for  15  months ;  afterwards  D  len* 
C  5  dollars  ;  how  long  ought  he  to  keep  it  to  satisfy  him  for 
tlie  former  favour  1 

101.  A  borrowed  of  B  17  dollars  for  11  months,  promis- 
ing him  a  like  kindness ;  afterwards  B  lent  A  25  dollars. 
How  long  ought  he  to  keep  it  ] 

Note.  Find  how^  long  he  ought  to  keep  1  dollar,  and  then 
how  .ong  he  ought  to  keep  25  dollars. 

102.  I  lent  a  friend  257  dollars,  which  he  kept  15  months, 
promising  to  do  me  a  like  kindness,  but  he  was  not  able  to 
let  me  have  more  than  100  dollars;  how  long  ought  I  to 
keep  it  1 

103.  A  owes  B  notes  to  be  paid  as  follows  :  7  dollars  to 
be  paid  in  3  months,  and  5  dollars  to  be  paid  in  8  months ; 
but  he  wishes  to  pay  the  whole  at  once.  In  what  time  ought 
be  to  pay  it  1 

Note.  7  dollars  for  3  months  is  the  same  as  1  dollar  for 
21  months  ;  and  5  dollars  for  8  months  is  the  same  as  1  dol- 
lar for  40  months.  40  +  21  =  61,  and  7  -f  5  =  12.  He 
might  have  1  dollar  61  months;  the  question  now  is  how 
long  he  may  keep  12  dollars.  It  is  evident  he  might  keep 
it  -^^  of  61  months. 

104.  C  owes  D  S380,  to  be  paid  as  follows ;  $100  in  6 
months ;  $120  in  7  months  ;  and  $160  in  10  months.     He 


Part  2.  ARITHMETIC-  223 

wishes  to  pay  the  whole  at  once.     In  how  long  a  time  ought 
he  to  pay  it  ? 

105.  A  merchai^t  has  due  to  him  300c£.  to  be  paid  as  fol- 
lows ;  50^.  in  2  months  ;  100c£  in  5  months  ;  and  the  rest 
in  8  months.  It  is  agreed  to  make  one  payment  of  the 
whole.     In  what  time  ought  he  to  receive  it  ? 

106.  F  owes  H  $1GH)0,  of  which  $200  is  to  be  paid  pre^ 
sent,  $400  in  5  months,  and  the  rest  in  15  months.  Tney 
agree  to  make  one  payn.^ent  of  the  whole.  Required  the 
time  ? 

107.  A  merchant  has  duo  a'  certain  sum  of  money,  of 
which  i  is  to  be  paid  in  2  mont  hs,  \  in  3  months,  and  the 
rest  in  6  months.  In  what  time  ought  he  to  receive  the 
whole  1 

108.  A  merchant  has  three  note.s  Me  to  him  as  follows  i 
one  of  $300  due  in  2  months  ;  one  o/  1^250  due  in  5  months; 
and  one  of  $180  due  3  months  ago  ;  ihe  whole  of  which  he 
wishes  to  receive  now.  What  ought  hO  to  receive,  allowing 
6  per  cent,  interest  1 

Note,  First  find  the  equated  time,  and  the.n  the  interest 
or  discount  for  present  payment,  as  shall  be  Tounvd  necessary. 

$300  for  2  months  =  1  dol.  for    600  n.^onths; 
$250  for  5  months  =  1  dol.  for  1250  mc^nths. 

1850 

The  two  notes  not  yet  due  are  the  same  as  1  dv.>llar  for 
1850  months.  But  he  has  had  $180  3  months  after  it  was 
duo,  which  is  the  same  as  1  dollar  for  540  months.  This 
must  be  taken  out  of  the  other,  and  there  will  remain  1  i^^y 
lar  for  1310  months.  If  he  can  have  1  dollar  for  13i'0 
months,  how  long  can  he  have  $730  1 
131,0  (73,0 

73       

1.8  nearly  =  1  month  and  24  days. 

580 

584 
As  it  is  not  due  until  1  month  and  24  days  after  thu 
time,  it  must  be  discounted  for  that  time.  See  Part  I. 
Art.  XXIV.,  example  130  and  following.  6  percent  for  1 
year  is  ^^  per  cent,  or  .009  for  1  month  and  24  days  The 
fraction  then  is  |^|.     $730  is  f5  o|  of  what  ? 


224  ARITHMETIC.  Part% 

100.  A  gave  B  four  notes  as  follows  ;  one  of  $75,  dated 
6th  June,  1819,  to  be  paid  in  4  months;  one  of  8150,  dated 
15th  August,  to  be  paid  in  6  months;  one  of  $170,  dated 
11th  September,  to  be  paid  in  5  months;  and  one  of  $300 
dated  15th  November,  to  be  paid  in  3  months.  They  were 
all  without  interest  until  they  were  due.  On  1st  January, 
18*20,  he  proposed  to  pay  the  whole.  What  ought  he  to 
pay? 

110.  A  owes  B  $158.33,  due  in  11  months  and  17  days, 
without  interest,  which  he  proposes  to  pay  at  present.  What 
ought  he  to  pay,  when  the  rate  of  money  is  5  per  cent.  ? 

Note.     The  rate  per  cent,   for  11    mo.  17  days,  at  5  per 
cent,  a  year,  is  about  4^^^  per  cent,  or  .048,  consequently  the 
amount  of  1  doll,  is  $1,048.     $158.33  is  f^f  of  the  num 
ber. 

It  is  easy  to  find  the  rate  per  cent,  of  the  discount  for  any 
given  time,  when  the  rate  of  interest  is  given.  When  interest 
is  6  per  cent.,  that  is,  y-^ q,  the  discount  is  yf  g-,  because  the  dis- 
count of  106  dolls,  is  6  dolls.  If  j^-g  be  converted  into  a  de- 
cimal, it  gives  the  rate  of  discount  in  decimals,  so  that  it  may 
be  computed  in  the  same  manner  as  interest.  This  changed 
to  a  decimal  is  .0566.  .057  —  is  sufficiently  exact.  This  is 
^h  percent  The  rate  must  be  found  for  the  time  required, 
before  it  is  changed  to  a  decimal. 

In  the  last  example  the  fraction  would  be  yjf  g,  which  is 
.046  nearly.  Multiply  the  sum  by  this,  and  you  will  have 
the  discount,  which  subtracted  from  the  sum,  will  be  the  an- 
swer required. 

111.  What  is  the  discount  of  $143.87  for  1  year  and  5 
months,  when  interest  is  6  per  cent.  1 

112.  What  is  the  present  worth  of  a  note  of  $84.67,  due 
in  1  year,  3  months,  and  14  days,  without  interest,  when  the 
rate  of  interest  is  S\  per  cei.t.  ? 

113.  A  man  has  a  note  of  $647  due  in  2  years  and  7 
months,  without  interest ;  but  being  in  want  of  the  money,  he 
sells  the  note  ;  what  ought  he  to  receive,  when  the  usual 
rate  of  interest  is  6  per  cent.  ? 

114.  A  gentleman  divided  $50  between  two  men,  A  and 
B.     A's  share  was  |  of  B's.     What  was  the  share  of  each  ? 

Note.  This  question  is  to  divide  the  number  50  into  two 
parts,  that  shall  be  in  the  proportion  of  3  and  7  ;  that  is,  one 


Part  "a,  ARITHMETIC.  225 

shall  have  3  as  often  as  the  other  shall  have  7.     7  -j-  3  rrr 
10.     A  had  V'^  and  B  ,V- 

115.  A  gentleman  bequeathed  an  estate  of  $12500  be- 
tween his  wife  and  son.  The  son's  share  was  \  of  the  share 
of  the  wife.     What  was  the  share  of  each  1 

116.  What  is  the  hour  of  the  day,  when  the  time  past 
from  midnight  is  equal  to  y\  of  the  time  to  noon  1 

117.  Two  men  talking  of  their  ages,  one  says  |  of  my 
age  is  equal  to  |  of  yours :  and  the  sum  of  our  ages  is  95. 
What  were  their  ages  1 

Note.  To  find  the  proportions,  reduce  them  to  a  commou 
denominator  and  take  the  numerators. 

118.  If  a  man  can  do  |  of  a  piece  of  work  in  one  day,  in 
what  part  of  a  day  can  he  do  }  of  it  ?  How  long  will  it  take 
him  to  do  the  whole  1 

1 19.  A  farmer  hired  two  men  to  mow  a  field  ;  one  of 
them  could  mow  i  of  it  in  a  day,  and  the  other  |  of  it.  What 
part  of  it  would  they  both  together  do  in  a  day  1  How  long 
would  it  take  them  both  to  mow  it  1 

120.  A  gentleman  hired  3  men  to  build  a  wall ;  the  first 
could  do  it  alone  in  8  days,  the  second  in  10  days,  and  the 
third  in  12  days.  What  part  of  it  could  each  do  in  a  day  ? 
How  long  would  it  take  them  all  together  to  finish  it  1 

121.  A  man  and  his  wife  found  that  when  they  were  to- 
gether, a  bushel  of  corn  would  last  15  days,  but  when  the 
man  was  absent,  it  would  last  the  woman  alone  27  days. 
What  part  of  it  did  both  together  consume  in  1  day  ?  What 
part  did  the  woman  alone  consume  ?  What  part  did  the  man 
alone  consume  1     How  long  would  it  last  the  man  alone  1 

122.  Three  men  lived  together,  one  of  them  found  he 
could  drink  a  barrel  of  cider  alone  in  4  weeks,  the  second 
could  drink  it  alone  in  6  weeks,  and  the  third  in  7  weeks. 
How  long  would  it  last  the  three  together  ? 

123.  A  cistern  has  3  cocks  to  fill  it,  and  one  to  empty  it. 
One  cock  will  fill  it  alone  in  3  hours,  the  second  in  5  hours, 
and  the  third  in  9  hours.  The  other  will  empty  it  in  7 
hours.  If  all  the  cocks  are  allowed  to  run  together,  in  what 
time  will  it  be  filled  ? 

124.  Divide  25  apples  between  two  persons,  so  as  to  give 
one  7  more  than  the  other. 


226  ARITHMETIC.  Part  % 

Note,  Give  one  of  them  7,  and  then  divide  the  rest 
equally. 

125.  A  gentleman  divided  an  estate  of  $15000  between 
his  two  sons,  giving  the  elder  $2500  more  than  the  younger. 
What  was  the  share  of  each  ? 

126.  A  gentleman  bequeathed  an  estate  of  $50000,  to  his 
wife,  son,  and  daughter;  to  his  wife  he  gave  $1500  more 
than  to  the  son,  and  to  the  son  $3500  more  than  to  the 
daughter.     Wiiat  was  the  share  of  each  ? 

127.  A,  B,  and  C,  built  a  house,  which  cost  $35000 ;  A 
paid  $500  more,  and  C  $300  less  than  B.  What  did  each 
pay? 

128.  A  man  bought  a  sheep,  a  cow,  and  an  ox,  for  $62  ; 
for  the  cow  he  gave  $10  more  than  for  the  sheep  ;  and  for 
the  ox  $10  more  than  for  both.     What  did  he  give  for  each  ? 

129.  A  man  sold  some  calves  and  some  sheep  for  $108; 
the  calves  at  $5,  and  the  sheep  at  $8  apiece.  There  were 
twice  as  many  calves  as  sheep.  What  was  the  number  of 
each  sort  1 

Note.     There  were  two  calves  and  one  sheep  for  every 

$18. 

130.  A  farmer  drove  to  market  some  oxen,  some  cows, 
and  some  sheep,  which  he  sold  for  $749  ;  the  oxen  at  $28, 
the  cows  at  $17,  and  the  sheep  at  $7.50.  There  were  twice 
as  many  cows  as  oxen,  and  three  times  as  many  sheep  as 
cows.     How  many  were  there  of  each  sort  ? 

131.  A  man  sold  16  bushels  of  rye,  and  12  bushels  of 
wheat  for  c£8.  16s.  The  wheat  at  3s.  per  bushel  more  than 
the  rye.     What  was  each  per  bushel  ? 

Note.  The  whole  of  the  wheat  came  to  36s.  more  than 
the  same  number  of  bushels  of  rye.  Take  out  36s.,  and  the 
remainder  will  be  the  price  of  28  bushels  of  rye. 

132.  Four  men,  A,  B,  C,  and  D,  bought  an  ox  for  $50, 

which  they  agreed  to  share  as  follows  :  A  and  B  were  to 
have  the  hind  quarters,  C  and  D  the  fore  quarters.  The 
hind  quarters  were  considered  worth  i  cent  per  lb.  more 
than  tlie  fore  quarters.  A's  quarter  weighed  217  lb.  ;  B's 
223  lb.  ;  C's  214  lb.  ;  and  D's  219  lb.  The  tallow  weigh- 
ed 73  lb.,  which  they  sold  at  8  cents  per  lb. ;  and  the  hide 
43  lb.,  which  they  sold  at  5  cents  per  lb.  What  ought 
each  to  pay  ? 


Part  2.  ARITHMETIC.  227 

133.  At  the  time  they  bought  the  a])ove  ox,  tlie  fore  quar- 
ters of  beef  were  worth  0  cents  per  lb.,  and  the  hind  quar- 
ters 6^-  cents  per  lb.  It  is  required  to  find  what  each  ought 
to  pay  in  this  proportion. 

Note.  This  is  a  more  just  manner  of  dividing  the  cost, 
than  that  in  the  last  example.  It  may  be  done  by  finding 
what  the  quarters  would  come  to,  at  this  rate,  and  then  di- 
viding the  real  cost  in  that  proportion. 

134.  Said  A  to  B,  my  horse  and  saddle  together  are 
worth  $150,  but  my  horse  is  worth  9  times  as  much  as  the 
saddle.     What  was  the  value  of  each  ? 

135.  A  man  driving  some  sheep  and  some  cattle,  being 
asked  how  many  he  had  of  each  sort,  said  he  had  174  in 
the  whole,  and  there  were  /„  ^^  many  cattle  as  sheep.  Re- 
quired the  number  of  each  sort. 

136.  A  man  driving  some  sheep,  and  some  cows,  and  some 
oxen,  being  asked  how  many  he  had  of  each  sort,  answered, 
that  he  had  twice  as  many  sheep  as  cows,  and  three  times 
as  many  cows  as  oxen  ;  and  that  the  whole  number  was  80. 
Required  the  number  of  each  sort. 

137.  A  gentleman  left  an  estate  of  $13000  to  his  four 
sons,  in  such  a  manner,  that  the  third  was  to  have  once  and 
one  half  as  much  as  the  fourth,  the  second  was  to  have  as 
much  as  the  third  and  fourth,  and  the  first  was  to  have  as 
much  as  the  other  three.     What  was  the  share  of  each  ? 

138.  A,  B,  and  C  playing  at  cards,  staked  324  crowns  ; 
but  disputing  about  the  tricks,  each  man  took  as  many 
crowns  as  he  could  get.  A  got  a  certain  number  ;  B  as 
many  as  A,  and  15  more  ;  and  C  \  part  of  both  their  sums 
added  together.     How  many  did  each  get  ? 

139.  The  stock  of  a  cotton  manufactory  is  divided  into  32 
shares,  and  owned  equally  by  8  persons,  A,  B,  C,  &c.  A 
sells  3  of  his  shares  to  a  ninth  person,  who  thus  becomes  a 
member  of  the  company,  and  B  sells  2  of  his  shares  to  the 
company,  who  pay  for  them  from  the  public  stock.  After 
this,  A  wishes  to  dispose  of  the  remainder  of  his  part.  What 
proportion  of  the  whole  stock  does  he  own  1 

140.  Three  persons,  A,  B,  and  C,  traded  in  company.  A 
put  in  $75  ;  B  $40  ;  and  C  a  sum  unknown.  They  gained 
164,  of  which  C  took  $18  for  his  share.  What  did  C  put 
ml 

141.  How  many  cubic  feet  in  a  cistern,  4  ft.  2  in.  long, 
3  ft.  8  in.  wide,  and  2  ft.  7  in.  high  ? 


^228  ARITHMETIC.  Part  2. 

A  method  of  doing  this  by  decimals  has  already  been 
shown.  It  is  now  proposed  to  do  it  by  a  method  called  duo 
decimals. 

First,  I  find  the  square  feet  in  the  bottom  of  the  cistern. 

4-2-  ft 


3  ft.  8  : 

in.  - 

=  3/jft. 

4  ft.  2  in.  z: 

tt 

2  ft.  7  in.  = 

■i-r 

tIt 

square  feet 

15ft  + 

-^ 

in  the  bottom. 

8«  + 
30ft  + 

+  T^a 

Ans.  39/j  -{-  yij  -f  -pj*^^  cubic  feet  in  the  cistern. 

I  say  y8_  of  JL  is  -1^6-  —  -1-  ^  _i_,  I  write  down  the  -^^^ 
and  reserve  the  yL  ;  then  -f^  of  4  is  f|  and  yV  (which  was 
reserved)  is  ff  =  2y^2'  which  I  write  down.  Then  3  times 
T2  ^s  tV'  ^"^  ^  times  4  are  12.  These  added  together  make 
15j-%-  +  T44  square  feet.  Then  to  find  the  cubic  feet,  I 
multiply  this  by  2^.  ^  of  ^^  is  ^f  f^  =  yfr  +  ttV^,  I 
write  the  y^,  and  reserve  the  j^ ;  then  y\  of  y^^  is  fJ^, 
and  y|^  (which  were  reserved)  are  f-^  =  i.  _|.  _li_  ;  I 
write  down  the  -^  and  reserve  the  y'^  5  then  y^^  of  15  are 
8^  and  yL.  (which  was  reserved)  is  Sff.  2  timci  y|^  are 
7^4  ;  and  2  times  y^2  are  y\,  and  2  times  15  are  30.  Adding 
them  together,  yf ^  and  yW  are  y^^:  =:  y^  -f-  y^-  ;  I  write 
the  j^, ,  and  reserve  the  y'^-  ;  then  ||  and  y^^  are  j|,  and  y'g 
(which  was  reserved)  is  \l  r=  ly^^.     The  whole  is  39 -\  -f 

7  4 

t^  .ve  know  that  12ths  multiplied  by  ISihs  will  pro 

duce  -^4ths,  and  that  y'^  make  -^r,  and,  also,  that  144ths 
multiplied  by  12ths  produce  1728ths,  and  that  yf|^  make 
,i4Tj  W6  may  write  the  fractions  without  their  denominators, 
if  we  make  some  mark  to  distinguish  one  from  the  other. 
It  is  usual  to  distinguish  12ths  by  an  accent,  thus  ('),  I44th3 
thus  ("),  1728ths  thus  ('"),  &c.  12ths  are  called  primes  ; 
I44ths  seconds  ;  I728ths  thirds,  &c 


PaH  2.  ARITIIMP:TIC. 

Operation. 
4    2' 
3    8' 


2 

9' 

4" 

12 

6' 

15 

3' 

4" 

2 

7' 

8 

10' 

11"  4'^' 

30 

6' 

8" 

Cubic  feet  39  5'  7"  4'" 
The  operation  is  precisely  the  same  as  before.  To  adopt 
the  language  suited  to  this  notation,  we  scaj,  units  multiplied 
hy  primes  or  primes  by  units  produce  primes,  seconds  by  units 
produce  seconds,  S^-c.  primes  by  primes  produce  seconds,  se- 
conds by  primes  produce  thirds.  Also  12  thirds  make  1 
second,  12  seconds  1  prime,  12  primes  make  1  foot,  whether 
long,  square,  or  cubic.  The  same  principle  extends  to  fourths, 
fifths^  Sfc. 

142.  How  much  wood  in  a  load  4  ft.  8  in.  high,  3  ft.  11 
in.  broad  and  8  ft.  long  ? 

Note.  Multiply  the  height  and  breadth  together,  and 
divide  by  2.     See  page  102. 

143.  How  many  square  feet  in  a  floor  16  ft.  8  in.  wide, 
and  18  ft.  5  in.  long  ? 

144.  How  much  wood  in  a  pile  4  ft.  wide,  3  ft.  8  in.  high, 
and  23  ft.  7  in.  long  ? 

145.  If  11  barrels  of  cider  will  buy  4  barrels  of  flour,  and 
7  barrels  of  flour  will  buy  40  barrels  of  apples  ;  what  will  1 
barre'  of  apples  be  worth,  when  cider  is  $2.50  per  barrel  1 

146.  A  person  buys  12  apples  and  6  pears  for  17  cents, 
and  afterwards  3  apples  and  12  pears  for  20  cents.  What  i« 
the  price  of  an  apple  and  of  a  pear  ? 

Note.  At  the  second  time  he  bought  3  apples  and  12 
pears  for  20  cents,  4  times  all  this  will  make  12  apples  and 
48  pears  for  80  cents  ;  the  price  of  l2  apples  and  6  pears 
being  taken  from  this,  will  leave  63  cents  for  42  pears,  which 
is  ^  I  cent  apiece 

20 


230  A  RITHMETIC.  Part  % 

147.  Two  persons  talking  of  their  ages,  one  says  |  of  mine 
is  equal  to  f  of  yours,  and  the  difference  of  our  ages  is  10 
years.     What  were  their  ages  ? 

148.  A  gentleman  divided  some  money  among  4  persons, 
giving  the  first  as  much  as  the  second  and  fourth  ;  the  se- 
cond as  much  as  the  third  and  fourth ;  the  third,  half  as 
much  as  the  first ;  and  the  fourth,  5  cents.  How  much  did 
he  give  to  each  ? 

149.  Two  persons,  A  and  B,  talking  of  their  ages,  A  says 
to  B,  I  of  mine  and  \  of  yours  are  equal  to  13 ;  B  says  to  A, 
\  of  mine  and  |  of  yours  are  equal  to  16.  What  was  the 
age  of  each  1 

150.  A  person  drew  two  prizes  ;  \  of  the  ^*rst,  and  \  of 
the  second  was  $120  ;  and  the  sum  of  the  two  was  $400. 
What  was  each  prize  1 

151.  Two  persons  purchase  a  house  for  $4200  ;  the  first 
could  pay  for  the  whole,  if  the  second  would  give  him  i  of 
his  money  ;  and  the  second  could  pay  for  the  whole,  if  the 
first  would  give  him  \  of  his  money.  How  much  money  had 
each. 

152.  A  man  bought  some  lemons  at  2  cents  each,  and  |-  as 
many,  at  3  cents  each,  and  then  sold  them  all  at  the  rate  of  5 
cents  for  2,  and  by  so  doing  gained  25  cents.  How  many 
lemons  did  he  buy  1 

153.  There  are  two  cisterns  which  receive  the  same  quan- 
tity of  water  ;  the  first  constantly  loses  \  of  what  it  receives ; 
after  running  7  days,  10  barrels  were  taken  from  the  second, 
and  then  the  quantity  of  water  in  the  two  was  equal.  How 
much  water  did  each  receive  per  day  ? 

154.  A  man  having  $100  spent  a  certain  part  of  it;  he 
afterwards  received  five  times  as  much  as  he  spent,  and  then 
his  money  was  double  what  it  was  at  first.  How  much  did 
he  spend  ? 

155.  A  man  left  his  estate  to  2  sons  and  3  daughters,  each 
son  had  5  dollars  as  often  as  each  daughter  had  4 ;  the  dif- 
ference between  the  sum  of  the  sons'  shares  and  that  of  the 
daughters,  was  $1000.     Required  the  share  of  a  son. 

156.  A  man  left  his  estate  to  his  wife,  son,  and  daughter, 
5*5  follows :  to  his  wife  -^  of  the  whole,  and  ^  as  much  as  the 
share  of  the  daughter  ;  to  his  son  \  of  the  whole,  and  to  the 
daughter  the  remainder,  which  was  $1000  less  than  the 
siiare  of  the  son.     What  was  the  share  of  each  1 

167    A  man  bought  some  oranges  for  25  cents  ;  if  he  had 


Part  %  ARITHMETIC.  231 

bought  3  less  for  tlie  same  money,  the  price  of  in  orange 
would  have  been  once  and  a  hal^*  of  the  price  he  gave. 
What  was  tlie  ])rice  of  an  orange  ? 

158.  A  man  divided  his  estate  among  his  children  as  fol- 
lows :  to  the  first  he  gave  twice  as  much  as  to  the  third,  and 
to  the  second  two  thirds  as  much  as  to  the  first ;  the  portion 
of  tlie  second  and  third  together  was  $1500.  What  was  the 
portion  of  each  1 

159.  A  man  bought  10  bushels  of  corn,  and  20  bushels  of 
rye  for  $30  ;  and  aTso  24  bushels  of  corn,  and  10  of  rye  for 
$27.     How  much  per  bushel  did  he  give  for  each  ? 

100.  A  man  travelling  from  Boston  to  Philadelphia,  a  dis- 
tance of  335  miles,  at  the  expiration  of  7  days,  found  that 
the  distance  which  he  had  to  travel  was  equal  to  |f  of  the 
distance  which  he  had  already  travelled.  How  many  miles 
per  day  did  he  travel '? 

101.'  A  man  left  his  estate  to  his  three  sons  ?  the  first  had 
$2000,  the  second  had  as  much  as  the  first,  and  i  as  much 
as  the  third,  and  the  third  as  much  as  the  other  two.  What 
was  the  share  of  each  ? 

102.  A  man  when  he  married  was  three  times  as  old  as  his 
wife  ;  15  years  afterwards  he  was  but  twice  as  old.  What 
was  the  age  of  each  when  they  were  married  ? 

103.  A  grocer  bought  a  cask  of  brandy,  ^  of  which  leaked 
out,  and  he  sold  the  remainder,  at  $1.80  per  gal.,  and  by 
that  means  received  for  it  as  much  as  he  gave.  How  much 
did  it  cost  him  per  gal.  1 

104.  A  and  B  laid  out  equal  sums  of  money  in  trade  ;  A 
gained  a  sum  equal  to  ^  of  his  stock,  and  B  lost  $225  ;  then 
A's  money  was  double  that  of  B.    What  did  each  lay  out  1 

105.  There  is  a  fish  whose  head  is  10  inches  long,  his 
tail  is  as  long  as  his  head  and  half  the  length  of  his  body, 
and  his  body  is  as  long  as  his  head  and  tail.  What  is  the 
length  of  the  fish  ? 

106  There  are  three  persons,  A,  B,  and  C,  whose  ages 
are  as  follows  :  A  is  20  years  old,  B  is  as  old  as  A  and  f  of 
the  age  of  C,  and  C  is  as  old  as  A  and  B  both.  What  are 
the  ages  of  B  and  C  ? 

107.  A  person  has  two  silver  cups  and  only  one  cover. 
The  first  cup  weighs  12  oz.  If  the  first  cup  be  covered,  il 
will  weigh  twice  as  much  as  the  second,  but  if  the  second 
cup  be  covered,  it  will  weigh  three  times  as  much  as  the 
first.  Required  the  weight  of  the  cover  and  of  the  second  cup. 


232  ARITHMETIC-  Part  2, 

168.  Three  persons  do  a  piece  of  wo)k  ;  the  first  and 
second  together  do  -i  of  it,  and  the  second  and  third  to- 
gether do  -J J.     What  part  of  it  is  done  by  the  second  ? 

169.  A  man  bought  apples,  at  5  cents  per  doz.,  half  of 
which  he  exchanged  for  pears,  at  the  rate  of  8  apples  for  5 
pears  ;  he  then  sold  all  his  apples  and  pears,  at  1  cent  each, 
and  by  so  doing  gained  19  cents.  How  many  apples  did  lie 
buy,  and  how  much  did  they  cost  ? 

170.  A  man  being  asKcd  the  hour  of  the  day,  answered 
ihat  it  was  between  7  and  8,  but  a  more  exact  answer  beino- 
required,  said  the  hour  and  minute  hands  were  exactly  to- 
gether.    Required  the  time. 

171.  What  is  the  hour  of  tlie  day  when  the  time  past 
from  noon  is  equal  to  -{j  of  the  time  to  midnight  7 

172.  What  is  the  hour  of  the  day  when  I  of  the  time  past 
from  midnight  is  equal  to  |  of  the  time  to  lioon  1 

173.  A  merchant  laid  out  $50  for  linen  and  cotton  cloth, 
buying  3  yards  of  linen  for  a  dollar,  and  5  yards  of  cotton 
for  a  dollar.  He  afterwards  sold  \  of  his  linen,  and  \  of  his 
cotton  for  $12,  which  was  60  cents  more  than  it  cost  him. 
How  many  yards  of  each  did  he  buy  ? 

174.  A  gentleman  divided  his  fortune  among  his  three 
soiis,  giving  A  8  as  often  as  B  5,  and  B  7  as  often  as  C  4  ; 
the  difference  between  the  shares  of  A  and  C  was  $7500, 
What  was  the  share  of  ea'^h  ? 

175.  A  tradesman  increased  his  estate  annually  by  $150 
more  than  the  fourth  part  of  it ;  at  the  end  of  3  years  it 
amounted  to  $1481 1/^.     What  was  it  at  first  1 

176.  A  hare  has  50  leaps  before  a  grey-hound,  and  takes 
4  leaps  to  his  3  ;  but  two  of  the  grey-hound's  leaps  are  equal 
to  3  of  the  hare's.  How  many  leaps  must  the  grey-hound 
take  to  overtake  the  hare  ? 

177.  A  labourer  was  hired  for  60  days,  upon  this  condition, 
that  for  every  day  he  worked  he  should  receive  $1.50  ;  and 
for  every  day  he  was  idle,  he  should  forfeit  $.50 ;  at  the  ex- 
piration of  the  time  he  received  $75.  How  many  days  did 
he  ^vork  ? 

178.  A  and  B  have  the  same  income,  A  saves  }  of  his, 
but  B,  by  spending  30^.  a  year  more  than  A,  at  the  end  of 
8  years  finds  himself  40^.  in  debt.  What  is  their  income, 
and  what  does  each  spend  per  year  ? 

179.  A  lion  of  bronze,  placed  upon  the  basin  of  a  foun- 
tain, can  spout  water  into  the  basin  through  his  throat,  his 


Pnrt^.  ARrTHMETlC.  233 

eyes,  and  his  riglit  foot.  If  he  spouts  through  his  throat  only, 
he  will  fill  the  hasin  in  (J  hours  :  if  through  his  right  eye 
only,  he  will  fill  it  in  '2  days  ;  if  through  his  h^ft  eye  only,  he 
will  fill  it  in  ii  days  ;  if  through  his  right  foot  only,  he  will 
fill  it  in  4  hours.  In  what  time  will  the  basin  be  filled  if  the 
water  flow  through  all  the  apertures  at  once  ? 

180.  A  player  commenced  play  with  a  certain  sum  of 
money  ;  at  the  first  game  he  doubled  his  money,  at  the  se- 
cond he  lost  10  shillings,  at  the  next  game  he  doubled  what 
he  then  had,  at  the  fourth  game  he  lost  20  shillings  ;  twice 
the  sum  he  then  had  was  as  much  less  than  200s.,  as  three 
times  the  sum  would  be  greater  than  200s.  Required  the 
sum  with  which  he  commenced  play. 

181.  What  is  the  circumference  of  a  wheel  of  which  the 
diameter  is  5  feet  ? 

The  circumference  of  a  circle  is  3.1410,  or  more  exactly 
3.14151V26  times  the  diameter. 

182.  What  is  the  diameter  of  a  wheel  of  which  the  cir- 
cumference is  17  feet  1 


A  parallelogram  is  a 
figure  with  four  sides  in 
which  the  opposite  sides 
are  parallel  or  equidistant  f        d  e        c 

throughout  their  whole  extent.  In  the  adjacent  figure  a  b  c 
D  is  a  parallelogram,  and  also  a  b  e  f.  a  b  e  f  is  a  rectan- 
gular parallelogram,  or  a  rectangle,  ard  is  measured  as  ex- 
plained page  79.  It  is  easy  to  see  that  a  b  c  d  is  equal  to  a  b 
E  F,  because  the  triangle  b  c  e  is  equal  to  a  d  f.  The  contents 
of  a  parallelogram,  then,  is  found  by  multiplying  the  length 
of  one  of  its  sides  as  a  b,  by  the  perpendicular  which  mea- 
sures the  distance  from  that  side  to  its  opposite,  as  p.  e. 

D  C 

The  triangle  a  is  half  the  pa- 
rallelogram A  B  c  D.  The  area 
of  a  triangle,  therefore,  will  be 
half  the  product  of  the  Ijase  a  b, 
by  the  perpendicul?r  c  e.  If  the  a  e  b 

perpendicular  should  fall  without  the  triangle  it  will  be  the 
same. 

To  find  the  area  of  any  irregular  figure,  divide  »t  into  tri- 
angles. 

20  ♦ 


234  ARITHMETIC.  Pari  % 

To  find  the  area  of  a  circle,  multiply  half  the  diameter  by 
half  the  circumference.  Or  multiply  half  the  diameter  into 
itself,  and  then  multiply  it  by  3.1415926. 

To  find  the  solid  contents  of  a  round  stick  of  timber,  find 
the  area  of  one  end,  and  multiply  it  by  the  length. 

If  a  round  or  a  square  stick  tapers  to  a  point,  it  contains 
just  ^  as  much  as  if  it  were  all  the  way  of  the  same  size 
as  at  the  largest  end.  If  the  stick  tapers  but  does  not  come 
to  a  point,  it  is  easy  to  find  when  it  would  come  to  a  point, 
and  what  it  would  then  contain,  and  then  to  find  the  contents 
of  the  part  supposed  to  be  added,  and  take  it  away  from  the 
whole. 

183.  What  is  the  area  of  a  parallelogram,  of  which  one  side 
is  13  feet,  and  the  perpendicular  7  feet  ? 

^^5.  91  square  feet. 

184.  How  much  land  is  in  a  triangular  field,  of  which  one 
side  is  28  rods,  an^  the  distance  from  the  angle  opposite  that 
side  to  that  side,  lo  rods  ? 

Ans.  210  sq,  rods,  or  1  acre  and  50  j'ods. 

185.  How  many  square  inches  in  a  circle,  the  diameter 
10  inches  1  Aiis.  78.54  -|-  in. 

186.  How  many  solid  feet  in  a  round  stick  of  timber  10 
inches  in  diameter  and  17  feet  long  1 

Ans.  9.272  -\-ft. 

187.  How  many  cubic  feet  of  water  will  a  round  cistern 
hold  which  is  3  ft.  in  diameter  at  the  bottom,  4  ft.  at  top,  and 
5  ft.  high  1  Ans.  48.433  fL 


Geographical  and  Astronomical  Questions, 

188.  The  diameter  of  the  earth  is  7911.73  miles  ;  what  is 
its  circumference  ? 

189.  The  earth  turns  round  once  in  24  hours ;  how  far 
are  the  inhabitants  at  the  equator  carried  each  hour  by  this 
motion  1 

190.  The  circumference  of  the  earth  is  divided  into  360 
degrees  ;  how  many  miles  in  a  degree  ? 

191.  How  many  degrses  does  the  earth  turn  in  1  hour  ? 

192.  How  many  minutes  of  a  degree  does  the  earth  turn 
in  1  minute  of  time  ? 


Part%  ARITHMETIC.  235 

19:5.  VVl)at  IS  the  diirerence  in  the  time  of  two  places 
whose  difference  of  loncritiide  is  'Z''\'  4:V  ! 

11)4.  The  longitude  of  Boston  is  71'^  4'  W.  of  Greenwich, 
Kntrland.  What  is  the  time  at  Greenwich  when  it  is  1 1  h. 
43  niin.  morn,  at  Boston  ? 

195.  The  long,  of  Philadelphia  is  75"  09'  W.,  that  of 
Rome  12°  29  E.  What  is  the  time  at  Philadelphia,  wiien 
at  Rome  iv  is  6  h.  27  min.  even.  ? 

190.  The  eartii  moves  round  the  sun  in  1  year,  in  an 
orbit  nearly  circular.  Its  distance  from  the  sun  is  about 
95,000,000  of  miles  ;  what  distance  does  the  earth  move 
every  hour  1 

197.  The  Idt.  of  Turk's  Island  is  21''  30'  in.  and  the  long. 
is  about  tlie  same  as  that  of  Boston.  The  lat.  of  Boston  is 
42°  23'  N.     IIow  manv  miles  apart  are  they  ? 

198.  The  mouth  of  the  Columbia  river  is  about  125^  W. 
long.,  and  Montreal  is  about  73^  W.  long.,  they  are  in  about 
the  same  lat.  A  degree  of  longitude  in  that  latitude  is  about 
48.3  miles.  How  many  miles  are  they  apart,  measuring  on 
a  parallel  of  latitude  1 


Examples  in  Exchange. 

It  is  not  necessary  to  give  rules  for  exchange.  There  are 
hooks  which  explain  the  relative  value  of  foreign  and  Ameri- 
can coin,  weights,  and  measures.  The  one  may  ue  exchang- 
ea  to  the  other  l>y  multiplication  or  division. 

199.  What  is  the  value  of  13^".  14s.  Sa.  English  or  sier- 
ling  money,  in  Federal  money  1 

It  will  be  most  convenient  to  reduce  the  shillings  a;  d 
pence  to  the  decimal  of  a  pound.  For  the  value,  see  the  tar 
ble. 

200.  What  is  the  value  of  $153.78  in  sterling  money  1 

201.  What  is  the  value  of  853  francs,  50  centimes,  in 
Federal  money  ? 

202.  What  is  the  value  of  $287.42,  in  French  money  1 

203.  What  is  the  value  of  523  Dutch  gelders  or  florins, 
at  40  cents  each,  in  Federal  money  1 

204.  What  is  the  value  of  $98.59  in  Dutch  gelders. 

205.  What  is  the  value  of  387  ducats  of  Naples,  at  $777| 
each,  in  Federal  money  ? 


2S6  ARITHMETIC.  PaH  2. 

Tables  of  Coin,  Weights,  and  Measures. 

Denominations  of  Federal  money  as  determined  by  an  Act 
of  Congress,  Aug.  8,  1786. 

10  mills  make    1  cent   marked   c. 

10  cents  1  dime  d. 

10  dimes  1  dollar  $ 

10  dollars  1  Eagle  E. 

The  coins  of  Federal  money  are  two  of  gold,  four  of  sil- 
ver, and  two  of  copper.  The  gold  coins  are  an  eagle  and 
half-eagle  ;  the  silver,  a  dollar,  half-dollar,  double-dime,  and 
dime  ;  the  copper,  a  cent  and  half-cent.  The  standard  gold 
and  silver  is  eleven  parts  fine,  and  one  part  alloy.  The 
weight  of  fine  gold  in  the  eagle  is  240.268  grains  ;  of  fine 
silver  in  the  dollar,  375.64  grains  ;  of  copper  in  100  cents, 
Sj-  lbs.  avoirdupois.* 

ENGLISH    MONEY. 

4  farthings  make  1  penny  d.  value  in  U.  S.  $0,019 

12  pence  1  shilling  s.  .228 

20  shillings  1  pound  £.  4.4444 

21  shillings  1  guinea  4.6724 

FRENCH    MONEr. 

100  centimes  make  1  franc,  value  $.1875. 

TROY    WEIGHT. 

24  grains  (gr.)  make  1  penny-weight    dwt. 
20  dwt.  1  ounce  oz. 

12  oz.  1  pound  lb. 

By  this  weight  are  weighed  jewels,  gold,  silver,  corflj 
bread,  and  liquors. 

apothecaries'  weight. 

20  grains  (gr.)  make  1  scruple  sc. 

3  sc.                           1  dram  dr.  or  3 

8  dr.                          1  ounce  oz.  or  | 

12  oz.  1  lb. 

•  The  above  are  the  coins  which  were  at  first  contemplated,  but  the 
do»ible-dimc  has  never  been  coined.  Twenty-fivc-cent  pinces  and 
half-dimes  nave  been  coined. 


Pfl»'/2.  ARITHMETIC.  237 

Apothecaries  use  this  weight  in  compounding  their  medi- 
cines ;  but  they  buy  and  sell  their  drugs  by  Avoirdupois 
weiglit.  Apothecaries'  is  the  same  as  Troy,  having  only  some 
dilierent  divisions. 


AVOIRDUPOIS    WEIGHT. 

6  drams  (dr.)  make  1  ounce  oz. 

16  oz.  1  pound  lb. 

28  lbs.  1  quarter  qr. 

4  qrs.  1  hundred-weight      cwt 

20  cwt.  1  ton  T. 

By  this  weight  are  weighed  all  things  of  a  coarse  and 
drossy  nature  ;  such  as  butter,  cheesp,  flesh,  grocery  wares, 
and  all  metals  except  gold  and  silver. 

DRY    MEASURE. 

2  pints  (pt.)  make  1  quart  qt. 

8  qts.  1  peck  pk. 

4  pks.  1  bushel  bu. 

8  bu  1  quarter  .\r. 

The  diameter  of  a  Winchester  bushel  is  \S^  inches,  and 
its  depth  8  inches. — And  one  gallon  by  dry  measure  con- 
tains 2<j8-|^  cubic  inches. 

By  this  measure  salt,  lead  ore,  oysters,  corn,  and  other  dry 
goods  are  measured. 


ALE    OR    BEER    MEASURE. 

2  pints  (pt.)  make  1  quart  qt 

4  qts.  1  gallon  gal. 

8  gals.  1  firkin  of  ale  fir. 

9  gals,                      1  firkin  of  beer  fir. 
2  fir.                         1  kilderkin  Xil. 

2  kil.  1  barrel  bar. 

3  kil.  1  hogshead  hhd. 
3  bar.                        1  butt                  butt. 

The  ale  gallon  contains  282  cubic  inches.  In  London 
the  ale  firkin  contains  8  gallons,  trnd  the  beer  firkin  9  ;  other 
measures  being  in  the  same  proportion. 


■2aS  ARITHMETIC.  Po^    2 


ARITHMETIC. 

F 

WINE 

MEASURE 

2  pints  (pt.) 

make  1  quart 

qt 

4qts 

1  gallon 

gal. 

42  gals. 

1  tierce 

tier. 

63  gals. 

1  hogshead 

hhd. 

84  gals. 

1  puncheon 

pun. 

2  hhds. 

1  pipe  or  butt 

p.  or  b. 

2  pipes 

1  tun 

T. 

18  gals. 

1  runlet 

run. 

31  J-  gallons 

1  barrel 

bar. 

The  wine  gallon  contains  231  cubic  inches. 
By  this  measure  brandy,  spirits,  perry,  cider,  mead,  vine- 
gar, and  oil  are  measured. 

CLOTH    MEASURE. 

Si  inches  make  1  nail  nl. 

4     nls.  1  quarter  qr. 

4  qrs.  1  yard  yd. 

3     qrs.  1  ell  Flemish  Ell  FL 

5  qrs.  I  ell  English  Ell  Eng. 
5    qrs.                1  aune  or  ell  French. 

The  French  aune  is  42  inches. 

LONG    MEASURE. 

3    barley  corns  make  1  inch  in. 

12    in.  1  foot  ft. 

3    ft.  1  yard  yd. 

5i  yds.  1  pole  or  rod        pole 

40    poles  1  furlong  fur. 

8    fur.  1  mile  ml. 

3    mis.  1  league  1. 

GO    geographical  miles,  or 
69|  statute  miles  1  degree  nearly,  deg.  <m*  ° 

360    degrees  the  circumference  of  the  earth. 

Also,  4  inches  make  1  hand 

5  feet  1  geometrical  pace 

6  feet  1  fathom 
6  points             1  line 

12  lines  1  inch 


Part  2. 


ARITHMETIC. 

SQUARE    MEASURE. 


144    inches  make  1  foot 
9    ft 


30i  yds.  or  ) 
272|  ft.  / 

40    poles 
4    roods 


239 


ft. 

]  yard  yd. 

1  pole,  rod,  or  perch. 

1  rood. 
1  acre. 


CUBIC    OR    SOLID    MEASURE. 


17iJ8  inches  make 
27  feet 

40  feet  of  round  timber,  or  > 
50  feet  of  hewn  timber        ) 
128  solid  feet 

TIME. 

60  seconds  make 
60  minutes 
24  hours 

7  days 

4  weeks 

13  months,  1  day,  and  G  hours  ) 
or  365  days,  6  hours  ] 

12  calendar  months 


1  foot  ft. 

1  yard. 

1  ton  or  load. 

1  cord  of  wood. 


1  minute  m. 

1  hour  h. 

1  day  d. 

1  week  w. 
1  month 

1  Julian  year  Y. 

1  year. 


The  true  length  of  the  solar  year  is  365  days,  5  hours,  48 
min.  57  seconds. 


Reflections  on  Mathematical  Reasom7ig, 


If  the  learner  has  studied  '  'le  preceding  pages  attentively, 
he  has  had  some  practice  in  mathematical  reasoning.  It 
may  now  be  pleasant,  as  well  as  useful,  to  give  some  atten- 
tion to  the  principles  of  it. 

By  attending  to  the  objects  around  us,  we  observe  two 
properties  by  which  they  are  capable  of  being  increased  or 
diminished,  viz.  in  number  and  extent. 

Whatever  is  susceptible  of  increase  and  diminution  is  thse 
object  of  mathematics. 

Arithmetic  is  the  science  of  numbers. 

All  individual  or  single  things  arc  naturally  subjects  of 
number.  Extent  of  all  kinds  is  also  made  a  subject  of  num- 
ber, though  at  first  view  it  would  seem  to  have  no  connexion 
with  it.  But  to  apply  number  to  extent,  it  is  necessary  to 
have  recourse  to  artificial  units.  If  we  wish  to  compare 
two  distances,  we  cannot  form  any  correct  idea  of  their 
relative  extent,  until  we  fix  upon  some  length  with  which 
we  are  familiar  as  a  measure.  This  measure  we  call  07ie 
or  a  unit.  We  then  compare  the  lengths,  by  finding  how* 
many  times  this  measure  is  contained  in  them.  By  this 
means  length  necomes  an  object  of  number.  We  use  dif- 
ferent units  for  different  purposes.  For  some  we  use  tlie 
inch,  for  others  the  foot,  the  yard,  the  rod,  the  mile,  &c. 

In  the  same  manner  we  have  artificial  units  for  surfaces, 
for  solids,  for  liquids,  for  weights,  for  time,  &lc.  And  in  all 
there  are  different  units  for  different  purposes. 

When  a  measure  is  assumed  as  a  unit,  all  smaller  mea- 
sures are  fractions  of  it.  If  the  foot  is  taken  for  the  unit, 
inches  are  fractions.  If  the  rod  is  the  unit,  yards,  feet,  and 
inches  are  fractions,  and  the  smaller,  being  fractions  of  the 
larger,  are  fractions  of  fractions.  It  may  be  remarked,  that 
all  parts  are  properly  units  of  a  lower  order.     As  we  say  sin- 


Part  2  ARITHMETIC.  241 

gle  things  are  units,  so  when  they  are  cat  into  parts,  these 
parts  are  single  things,  and  consequently  units,  and  they  are 
numbered  as  such.  When  a  thing  is  divided  into  eight  equal 
parts,  for  example,  the  parts  are  numbered,  one,  two,  three, 
&LC.  As  we  put  together  several  units  and  make  a  collec 
tion  which  is  called  a  unit  of  a  higher  order,  so  any  single 
thing  may  be  considered  as  a  collection  of  parts,  and  these 
parts  will  be  units  of  a  lower  order.  The  unit  may  be  con- 
sidered as  a  collection  of  tenths,  the  tenths  as  a  collection  of 
hundredths,  &,c. 

The  first  knowledge  we  have  of  numbers  and  their  uses  iS 
derived  from  external  objects  ;  and  in  all  their  practical  uses 
they  are  applied  to  external  objects.  In  this  form  they  are 
called  concrete  numbers.  Three  horses,  five  feet,  seven  dol- 
lars, &/C.  are  concrete  numbers. 

When  we  become  familiar  with  numbers,  we  are  able  to 
think  of  them  and  reason  upon  them  without  reference  to 
any  particular  object,  as  three,  five,  seven,  four  times  three 
are  twelve,  &,c.     These  are  called  abstract  numbers. 

Though  all  arithmetic  operations  are  actually  performed 
on  abstract  numbers,  yet  it  is  generally  much  easier  to  reason 
upon  concrete  numbers,  because  a  reference  to  sensible  ob- 
jects shows  at  once  the  purpose  to  be  obtained,  and  at  the 
same  time,  suggests  the  means  to  arrive  at  it,  and  shows  also 
how  the  result  is  to  be  interpreted. 

Success  in  reasoning  depends  very  much  upon  the  perfec- 
tions of  the  language  which  is  applied  to  the  subject,  and 
also  upon  the  choice  of  the  words  which  are  to  be  used. 
The  choice  of  words  again  depends  chiefly  on  the  knowledge 
of  their  true  import.  There  is  no  subject  on  which  the  lan- 
guage is  so  perfect  as  that  of  mathematics.  Yet  even  in  this 
there  is  grf>at  danger  of  being  led  into  errors  and  difficulties, 
for  want  of  a  perfect  knowledge  of  the  import  of  its  terms. 
There  is  not  much  danger  in  reasoning  on  concrete  num- 
bers ;  but  in  abstract  numbers  persons  pretty  well  skilled  m 
mathematics,  are  sometimes  led  into  a  perfect  paradox,  and 
cannot  discover  the  cause  of  it,  when  perhaps  a  single  word 
would  remove  the  whole  difficulty.  This  usually  happens  in 
reasoning  from  general  princi})les,  or  in  deriving  particular 
consequences  fiom  them.  The  reason  is,  the  general  prin- 
ciples are  but  partially  understood.  This  is  to  be  attributed 
chiefly  to  the  manner  in  which  mathematics  are  treated  in 
most  elementary  books,  where  one  general  principle  is  buiU 
21 


'24^i  ARITHMETIC.  Part  2. 

upon  anolher,  without  bringing  into  view  the  particulars  on 
whicii  I  hey  are  actiinlly  founded. 

There  are  several  different  forms  in  which  subtraction 
may  appear,  as  may  be  seen  by  referring  to  Art.  VIII.  In 
order  to  employ  the  word  subtraction  in  general  reasoning, 
either  ot  the  operations  ought  readily  to  bring  this  word  to 
mind,  and  the  word  ought  to  suggest  either  of  the  operations. 

The  word  division  would  naturally  suggest  but  one  pur- 
pose^ mat  is,  to  divide  a  number  into  parts  ;  but  it  is  applied 
to  anoiuer  purpose,  which  apparently  has  no  immediate  con- 
nexion With  it,  viz.  to  discover  how  many  times  one  number 
is  contained  in  another.  In  fractions  the  terms  multi|)lica- 
tion  and  division  are  applied  to  operations,  which  neither  of, 
the  terms  would  naturally  suggest.  The  process  of  multiply- 
ing a  whole  number  by  a  fraction  (Art.  XVI.)  is  so  differ- 
ent from  what  is  called  multiplication  of  whole  numbers, 
that  it  requires  a  course  of  reasoning  to  show  the  connexion, 
and  much  practice,  to  render  the  term  familiar  to  this  opera- 
tion. These  remarks  apply  to  many  other  instances,  but 
they  apply  with  much  greater  force  to  the  division  of  whole 
numbers  by  fractions.  Arts.  XXIII.  and  XXIV.  are  in 
stances  ot  this.  It  is  difficult  to  conceive  that  either  of 
these,  and  more  especially  the  latter,  is  any  thing  like  divi- 
sion ;  and  it  is  still  more  difficult  to  conceive  that  the  opera- 
lions  in  these  two  articles  come  under  the  same  name.  When 
a  person  learns  division  of  whole  numbers  by  fractions  from 
general  principles,  where  neither  of  these  operations  is 
brought  into  view,  it  is  easy  to  conceive  how  very  imperfect 
his  idea  of  it  will  be.  The  truth  is,  (and  I  have  seen  nu- 
merous instances  of  it,)  that  if  he  happens  to  meet  with  a 
practical  case  like  those;  in  the  articles  mentioned  above,  any 
other  term  in  the  world  would  be  as  likely  to  occur  to  him 
as  division.  In  an  abstract  example  the  difficulty  would  be 
"very  much  increased. 

The  above  observations  suggest  one  practical  result, 
which  will  apply  to  mathematics  generally,  and  it  will  be 
found  to  apply  with  equal  force  to  every  other  subject.  In 
adopting  any  general  term  or  expression,  we  should  be  care- 
ful to  examine  it  in  as  many  ways  as  possible.  Secondly, 
we  should  be  careful  not  to  use  it  in  any  sense  in  which  we 
have  not  examined  it.  Thirdly,  if  we  find  any  difficulty  in 
uaing  it  in  a  case  where  we  are  sure  it  ought  to  apply,  it  is 
an  indication  that  we  do  not  fully  understand  it  in  that 
ecnse.  and  that  it  requires  further  examination. 


t*art  2  ARITHMETIC.  943 

I  shall  give  a  few  instances  of  errors  and  difficulties  into 
which  i)ersons,  not  suHiciently  acquainted  with  ihe  princi- 
ples, sometimes  fall. 

Suppose  a  person  has  obtained  a  knowledge  of  the  rule 
of  division  by  a  course  of  abstract  reasoning,  and  that  the 
only  definite  idea  that  he  attaches--  to  it  is,  that  it  is  the  op,/0- 
site  of  multiplication,  or  that  it  is  used  to  divide  a  number 
into  parts.  Let  him  pursue  his  arithmetic  in  this  way,  and 
learn  to  divide  a  vvhole  number  by  a  fraction.  lie  will  be 
astonished  to  find  a  quotient  larger  than  the  dividend ;  and 
if  the  divisor  be  a  decimal,  his  astonishment  will  be  still 
greater,  because  the  reason  is  not  so  obvious.  Let  him  di- 
vide 40  by  i  according  to  the  rule,  and  he  will  find  a  quo- 
tient IKK  Or  let  him  divide  45  by  .03  and  he  will  find  a 
quotient  1500.  This  seems  a  perfect  paradox,  and  he  will 
be  quite  unable  to  account  for  it.  Now  if  he  had  the  idea 
intimately  joined  with  the  term  division,  that  the  quotient 
shows  how  many  times  the  divisor  is  contained  in  the  divi- 
dend ;  and  also  a  proper  idea  of  a  fraction,  that  it  is  less 
than  ont%  instead  of  saying,  divide  40  by  1--,,  or  45  by  .03,  he 
would  say,  how  many  times  is  *  contained  in  40,  or  .03  in 
45  ;  and  all  the  difficulty  would  vanish. 

Innumerable  instances  occur,  which  show  the  iniporlance 
of  a  sinale  idea  attached  to  a  general  term,  which  the  term 
itself  would  not  readily  bring  to  mind,  but  which  a  single 
word  is  often  sufficient  to  recal.  The  most  important  acces- 
sory ideas  to  be  attached  to  the  term  division  are.  that  the 
quotient  shows  how  many  times  the  divisor  is  cotuaincd  in 
the  dividend  ;  and  that  it  is  the  reverse  of  multiplication. 
Those  for  subtraction  are  that  it  shows  the  diif'nrmcc  o^  the 
two  numbers  ;  and  that  it  is  the  reverse  of  addition. 

Sometimes,  it  is  asked  if  dollars  aid  pounds,  or  gallons 
be  multi|)lied  together,  what  will  they  produce  ?  If  dollars 
be  divided  bv  dollars,  what  will  they  produce  1  If  dollars 
be  divided  by  bushels,  what  will  they  produce  ?  &:c. 

It  is  observed,  in  square  measure,  that  the  length  multi 
piled  by  the  breadth  gives  the  number  of  square  feet  in  any 
rectangular  surface.  It  is  sometimes  asked,  if  dollars  be 
multiplied  by  dollars,  what  will  be  produced  ?  If  os.  3d.  be 
multiplied  by  3s.  8d.,  what  will  be  the  result  1 

It  is  observed  in  fractions,  that  tenths  divided  by  tenths, 
hundredths  by  hundredths,  &c.  produce  un'ts ;  from  this 
some  have  concluded,  that  a  cent  divided  by  a  cent,  or  a 


244  ARITHMETIC.  J*art  % 

mill  by  a  mill,  would  produce  a  dollar,  and  though  they  are 
aware  of  the  absurdity,  cannot  tell  how  to  avoid  the  conclu- 
sion. 

The  alxjve  difficulties  arise  chiefly  from  not  making  a 
proper  distinction  between  abstract  and  concrete  members. 
Not  one  of  these  cases  can  ever  occur  in  the  manner  here 
proposed.  They  are  imperfect  examples.  When  a  perfect 
examj)le  is  proposed,  which  involves  one  of  the  above  cases, 
the  difHculty  is  entirely  removed. 

It  IS  not  proper  to  speak  of  dollars  being  multiplied  r 
divided  b)  dollars  or  gallons. 

At  5  dollars  per  barrel,  what  cost  3  barrels  of  flour  1 

Instead  of  saying  that  5  dollars  is  to  be  multiplied  by  S 
barrels,  say  3  barrels  will  cost  three  times  as  much  as  1  bar- 
rel, that  is  three  times  5  dollars. 

If  1  dollar  will  buy  7  lbs.  of  raisins,  how  many  pounds 
may  be  bought  for  4  dollars  1 

Say  4  dolkrs  will  buy  4  times  as  many  pounds  as  i  dol- 
lar. In  these  two  examples  there  is  no  doubt  what  the  an- 
swer should  be.  In  one  it  is  dollars,  and  in  the  other  it  is 
pounds. 

In  a  piece  of  cloth  5  feet  long  and  3  feet  wide,  how  many 
square  feet  ? 

If  it  were  5  feet  long  and  1  foot  wide,  it  would  contain  5 
square  feet,  but  being  3  feet  wide  it  will  contain  three  times 
as  many,  or  three  times  5  feet. 

In  a  certain  town  a  tax  was  laid  of  1  dollar  upon  every 
$150;  how  much  did  a  man  possess  whose  tax  was  3  dol- 
lars? 

It  is  evident  that  he  possessed  three  times  $150. 

At  1  cent  each,  how  many  apples  may  be  bought  for  1 
cent  ? 

Here  the  divisor  is  1  cent  and  the  dividend  is  1  cent,  and 
the  result  is  an  apple  instead  of  a  dollar. 

How  many  gallons  of  wine  at  2  dollars  per  gal.,  may  be 
bought  for  H  dollars  7 

As  many  times  as  2  dollars  are  contained  in  6  dollars,  so 
many  gallons  may  be  bought. 

The  truth  is,  the  numbers  are  always  used  as  abstract 
numbers,  but  a  reference  to  particular  objects  is  kept  in 
view,  and  the  nature  of  the  question  will  always  show  to 
what  the  result  must  be  applied. 

It  may  however  be  established  as  a  general  principle,  that 


Parti,  ARITHMETIC.  245 

the  multiplier  and  multiplicand  are  never  applied  to  the 
same  object,  and  m  j)rncisely  the  same  way  ;  and  the  pro- 
duct will  be  ap|)!ied  to  the  object  which  is  mentioned  in  one 
denomination,  as  beinj/  the  value  of  a  unit  in  the  other. 

In  division  there  are  two  numbers  given  to  find  a  third, 
two  of  which  will  always  be  of  the  same  denomination,  and 
the  other  ditrerent,  or  differently  applied. 

If  the  divisor  and  dividend  are  oi' the  same  denomination 
and  applied  in  the  same  way,  the  question  is,  to  find  how 
many  limes  the  one  is  contained  in  the  other,  and  the  quo- 
tient will  be  applied  difierently. 

If  the  divisor  and  the  dividend  are  of  different  denomina- 
tions, or  differently  ai)plied  to  the  same  denomination,  the 
questio!!  is  to  divide  the  dividend  into  parts,  and  the  quo- 
tient will  be  ap|)lie(l  in  tlie  same  manner  as  the  di\idend. 

When  any  difficulty  occurs  in  solv.ig  a  question,  it  is  best 
to  supply  very  small  numbers,  and  solve  it  first  with  them, 
and  then  with  the  imml)ers  given.  If  the  question  is  in  an 
abstract  form,  endeavour  to  loruj  a  practical  one,  which  shall 
require  the  same  operation,  and  the  difficulty  is  generally 
very  much  diminished. 

In  all  cases  reason  from  many  to  one,  or  from  a  part  to 
one  ;  and  then  from  one  to  many  or  to  a  part.  If  several 
parts  be  given,  always  reason  from  them  to  one  part,  and 
then  to  many  parts,  or  to  the  whole. 


IMPROVED 

SCHOOL  BOOKS. 


Colburn^s  First  Lessons,  or,  Intellectual  Arithmetico 

The  merits  of  this  little  work  are  so  well  known,  and 
so  highly  appreciated  in  Boston  and  its  vicinity,  that 
any  recommendation  of  it  is  unnecessary,  except  to 
those  parents  and  teachers  in  the  country,  to  whom  it 
has  not  been  introduced.  To  such  it  may  be  interest- 
ing and  important  to  be  informed,  that  the  system  of 
whicJi  this  work  gives  the  elementary  principles,  is  found- 
ed on  this  simple  maxim  ;  that,  children  slwdd  be  instntrt- 
ed  in  every  science,  just  so  fast  as  they  can  understand  it. 
In  conformity  with  this  principle,  the  book  commences 
with  examples  so  simple,  that  they  can  be  perfectly 
comprehended  and  performed  mentally  by  children  of 
four  or  five  years  of  age  ;  having  performed  these,  the 
scholar  will  be  enabled  to  answer  the  more  difficult  ques- 
tions which  tbllow.  He  will  find,  at  every  stage  of  his 
progress,  that  what  he  has  already  done  has  perfectly 
prepared  him  for  what  is  at  present  required.  This 
will  encourage  him  to  proceed,  and  will  afford  him  a 
satisfaction  in  his  study,  which  can  never  be  enjoyed 
while  performing  the  merely  mec^hanical  operation  of 
ciphering  according  to  artificial  rules. 

This  method  entirely  supersedes  the  necessity  of  any 
rules,  and  the  book  contains  none.  The  scholar  learns 
to  reason  correctly  respecting  all  combinations  of  num- 
bers ;  and  il  he  reason.?  correctly,  lie  must  obtain  the 
desired  result.     The  scholar,  who  can  be  made  to  un- 


Improved  School  Books. 

dersland  how  a  sum  should  be  done,  needs  neither  book 
nor  instructer  to  dictate  how  it  must  be  done. 

This  admirable  elementary  Arithmetic  introduces  the 
scholar  at  once  to  that  simple,  practical  system,  which 
accords  with  the  natural  operations  of  the  human  mind. 
All  that  is  learned  in  this  way  is  precisely  what  will 
be  found  essential  in  transacting  the  ordinary  business 
of  life,  and  it  prepares  the  way,  in  the  best  possible 
manner,  for  the  more  abstruse  investigations  which  be- 
long to  maturer  age.  Children  of  five  or  six  years  of 
age  will  be  able  to  make  considerable  progress  in  the 
science  of  numbers  by  pursuing  this  simple  method  of 
studying  it ;  and  it  will  uniformly  be  found  that  this  is 
one  of  the  most  useful  and  interesting  sciences  upon 
whicli  their  minds  can  be  occupied.  By  using  this  work 
children  may  be  farther  advanced  at  the  age  of  nine 
or  ten,  than  they  can  be  at  the  age  of  fourteen  or  fifteen 
by  the  common  method.  Those  who  have  used  it,  and 
are  regarded  as  competent  judges,  have  uniformly  de- 
cided that  more  can  be  learned  from  it  in  one  year,  than 
can  be  acquired  in  two  years  from  any  other  treatise 
ever  published  in  America.  Those  who  regard  econo- 
my in  time  and  money,  cannot  fail  of  holding  a  work 
in  high  estimation  which  will  afford  these  important 
advantages. 

Colburn's  First  Lessons  are  accompanied  with  such 
mstructions  as  to  the  proper  mode  of  using  them,  as 
will  relieve  parents  and  teachers  from  any  embarrass- 
ment. The  sale  of  the  work  has  been  so  extensive,  that 
the  publishers  have  been  enabled  so  to  reduce  its  price, 
that  it  is,  at  once,  the  cheapest  £ind  the  best  Arithmetic 
m  the  country. 


Colhurnh  Sequel. 

This  work  consists  of  two  parts,  in  the  first  of  which 
the  author  has  given  a  great  variety  of  questions,  ar- 


Improved  School  Books, 

ranged  according  to  the  method  pursued  in  the  First 
Lessons  ;  the  second  part  consists  of  a  few  questions, 
with  the  solution  of  them,  and  such  copious  illustrations 
of  the  principles  involved  in  the  examples  in  the  first  part 
of  the  work,  that  the  whole  is  rendered  perfectly  intel- 
ligible. The  two  parts  are  designed  to  be  studied  to- 
gether. The  answers  to  the  questions  in  the  first  pan 
are  given  in  a  Key,  which  is  published  separately  for 
the  use  of  instructers.  If  the  scholar  find  any  sum 
difficult,  he  must  turn  to  the  principles  and  illustrations, 
given  in  the  second  part,  and  these  will  furnish  all  the 
assistance  that  is  needed. 

The  design  of  this  arrangement  is  to  make  the  scho- 
lar understand  his  subject  thoroughly,  instead  of  per- 
forming his  sums  by  rule. 

The  First  Lessons  contain  only  examples  of  num- 
bers so  small,  that  they  can  be  solved  without  the  use  of 
a  slate.  The  Sequel  commences  with  small  and  sunple 
combinations,  and  proceeds  gradually  to  the  more  exten- 
sive and  varied,  and  the  scholar  will  rarely  have  occa- 
sion for  a  principle  in  arithmetic,  which  is  not  fully 
illustrated  in  this  work. 


Colhurn\s  Introduction  to  Algebra, 

Those  who  are  competent  to  decide  on  the  merits 
of  this  work,  consider  it  equal,  at  least,  to  either  of  thd 
others  composed  by  the  same  author. 

The  publishers  cannot  desire  that  it  should  have  a 
higher  commendation.  The  science  of  Algebra  is  so 
much  simplified,  that  children  may  proceed  with  ease 
and  advantage  to  the  study  of  it,  as  soon  as  they  have 
finished  the  preceding  treatises  on  arithmetic.  The 
same  method  is  pursued  in  this  as  in  the  author's  other 
works ;.  every  thing  is  made  plain  as  he  proceeds  with 
his  subject. 

The  uses  which  are  performed  by  this  science,  give 
it  a  high  claim  to  more  general  attention.     Few  of  tJie 


Improved  School  Books, 

more  aostract  mathematical  investigations  can  he  con- 
ducted without  it ;  and  a  great  [jroportion  of  those,  for 
which  arithmetic  is  used,  would  be  performed  with 
much  greater  facility  and  accuracy  by  an  algebraic 
process. 

The  study  of  Algebra  is  singularly  adapted  to  disci- 
pline the  mind,  and  give  it  direct  and  simple  modes  of 
rea-soning,  and  it  is  universally  regarded  as  one  of  the 
most  pleasing  studies  in  which  the  mind  can  be  en- 
gaged. 

The  Author's  Preface, 

The  first  object  of  the  author  of  the  following  trea- 
tise has  been  to  make  the  transition  from  arithmetic 
to  algebra  as  gradual  as  possible.  The  book,  there- 
fore, commences  with  practical  questions  in  simple  equa- 
tions, such  as  the  learner  might  readily  solve  with- 
out the  aid  of  algebra.  This  requires  the  explanation 
of  only  the  signs  plus  and  minus,  the  mode  of  express- 
ing multiplication  and  division,  and  the  sign  of  equal- 
ity ;  together  with  the  use  of  a  letter  to  express  the  un- 
known quantity.  These  may  be  understood  by  any  one 
who  has  a  tolerable  knowledge  of  arithmetic.  All  of 
them,  except  the  use  of  the  letter,  have  been  explained 
in  arithmetic.  To  reduce  such  an  equation,  requires 
only  the  application  of  the  ordinary  rules  of  arithmetic  ; 
and  these  are  applied  so  simply,  that  scarcely  any  one 
can  mistake  them,  if  left  entirely  to  himself.  One  or 
two  questions  are  solved  first  with  little  explanation  in 
order  to  give  the  learner  an  idea  of  what  is  wanted,  and 
he  is  then  left  to  solve  several  by  himself. 

The  most  simple  combinations  are  given  first,  then 
those  which  are  more  difficult.  The  learner  is  expected 
to  derive  most  of  his  knowledge  by  sol  ^ing  the  exam- 
ples himself;  therefore  care  has  been  taken  to  make 
the  explanations  as  few  and  as  brief  as  is  consistent  with 
giving  an  idea  of  what  is  required. 

In  order  to  study  this  work  to  advantage,  the  learner 
should  solve  every  question  in  course,  and  do  it  algebnj^ 


Improved  School  Booh. 

icaJly.  If  he  finds  a  question  which  he  can  solve  as  easi- 
ly without  the  aid  of  algebra  as  with  it,  he  may  be  as- 
sured, this  is  what  the  author  expected.  If  he  first 
solves  a  question,  which  involves  no  difliculty,  he  will 
understand  perfectly  what  he  is  about,  and  he  will  there- 
by be  enabled  to  encounter  those  which  are  difficult. 

When  the  learner  is  directed  to  turn  back  and  do  in 
a  new  way,  something  he  has  done  before,  let  him  not 
fail  to  do  it,  for  it  will  be  necessary  to  his  future  pro- 
gress ;  and  it  will  be  much  better  to  trace  the  new  prin* 
ciple  in  what  he  has  done  before  than  to  have  a  new 
example  for  it. 

The  author  has  heard  it  objected  to  his  arithmetics 
by  some,  that  they  are  too  easy.  Perhaps  the  same  ob- 
jection will  be  made  to  this  treatise  on  algebra.  But 
in  both  cases,  if  they  are  too  easy,  it  is  the  fault  of  iH 
subject,  and  not  of  the  book.  For  in  the  First  Lessons, 
there  is  no  explanation  ;  and  in  the  Sequel  there  is 
probably  less  than  in  any  other  books,  which  explain  at 
all.  As  easy  however  as  they  are,  the  author  believes 
that  whoever  undertakes  to  teach  them,  will  find  the 
intellects  of  his  scholars  more  exercised  in  studying 
them,  than  in  studying  the  most  difficult  treatise  he  can 
put  into  their  hands. 


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